This question gets tackled frequently by various Catholic thinkers, bloggers, and apologists, and generally the answer is an unqualified "no". Indeed, frequently philosophically-inclined Catholics (and other Christians) express anger at those who say things like "now God has another little angel" in the context of a child's funeral. A conversation I had yesterday at a philosophy conference led me to mull this question over a bit, and my answer to this question is a qualified "yes". For this reason too, I don't think that people who say these things about the dead being angels should be corrected.
Part of this issue is the question of what is meant by 'angel'. If by 'angel' one means a person who is necessarily immaterial--that is, a person who cannot have or be a body--then of course it is impossible for a human person to be an angel. You have or are (again, it depends on what you mean by 'have' and 'are' which is the correct verb) a body right now. So it's impossible for you to be unable to have or be a body. So in that sense you can't be an angel. Likewise, if my angel you mean something definitely non-human, then you can't be an angel in that sense (at least on my view: there are some Christian thinkers who think, with good reasons, that we can exist, after death, as non-human persons.)
I think there are other senses of the term 'angel', however. Many in Catholic tradition (e.g. St. Thomas Aquinas) identified the angels with the "intelligences" of Greek philosophy (though this was opposed, even in the middle ages, by some, such as many Franciscan thinkers e.g. Peter John Olivi). Intelligences must be understood in terms of the hierarchy of beings. At the bottom of this hierarchy are purely material things. Next, there are vegetative organisms (e.g. plants) that can grow, but cannot perform cognitive acts and do not have conscious appetites--that is, feelings or conscious desires. Above that are animals, which are capable of sense perception: they can take in information about the particular things in their surroundings, and appetitively respond. Each level in this hierarchy is distinguished by a new, more unified, more powerful, more subject-like kind of form--the immaterial thing in a substance (an individual existing thing that is not an attribute of something) that causes that substance to be the kind of substance that it is. At the top of this hierarchy (under God) are the intelligences: substances capable of intellectual and free activity--which are activities that transcend the material, as a result of which the intelligences are pure forms, that is, purely immaterial beings.
Humans occupy a curious position in this hierarchy. On the one hand, we are animals, material beings capable of sense perception and needing matter for most of our activities (which, as with all substances, we get from our form--which we more usually call our "soul"). But on the other hand, we are capable of intellectual and free activities, which we perform just with our soul (albeit in connection with the body, except in the case of certain mystical experiences). On the one hand, we come into existence with our bodies, and our souls are the individuals that they are through being oriented to forming a particular bit of matter. But on the other hand, our souls can exist without our bodies, and they are that which gives our bodies their peculiar personal and spiritual existence--that is, they are that which makes our bodies not merely material or even only biological things, but rather things that fully share in and express our spiritual and intellectual lives.
On Aquinas' view, for example, this means that our bodies do pre-exist our bodies in a certain, qualified sense: not that they exist at earlier moment of time than the moment of our conception, but that, in the order of explanation, existence is first given to our souls, and then our souls give that existence to our bodies. In the fact that our souls first and foremost are what exist in us, and our bodies only exist by sharing in our soul's existence, we are utterly unlike the other animals. It is for these reasons that some 20th century Thomists (such as Anton Pegis and Karol Wojtyla--better known as St. John Paul II) called human persons "incarnated intelligences" or "spiritualized bodies".
Aquinas goes so far as to say that our souls are the same kind (or, more precisely, the same genus) of thing as the angels--both our souls and the angels are intellectual substances. But in one of my academic writings (see there also for references for the above material from Aquinas) I've argued that we have good reason to go further. The issue I was discussing there was whether we can say that the souls of dead human persons are still persons, which is a big debate in current Christian philosophy. When my Uncle Don recently died, his soul went onto heaven, hell, or purgatory. Can we say of that soul that it is a person? Can we say that it is Uncle Don--that is, can we say that Uncle Don is right now in heaven, hell, and purgatory?
As I read him, Aquinas answers these questions "no" (a position called "corruptionism": the human person entirely "corrupts" or goes out of existence at death): to be a human person is to be a rational animal, and that's to have a body. No body, no person. We might pray to some of the not-yet-resurrected dead by name (like when we say "St. Peter, pray for us"). But Aquinas thinks we're really just praying to St. Peter's soul, which is not literally St. Peter, though it still has St. Peter's thoughts and virtues, and is enjoying God in the beatific vision. St. Peter won't exist again until the Resurrection.
This is a respectable position. It's one thing that motivates the ire of those who say that we should not call the dead "angels". It's motivated by a desire to maintain that we are animals, and to emphasize the importance of the Resurrection of the body. You won't be there without your body. This position was held, I think, by nearly every Western Christian thinker at least from the 11th to the 19th centuries, and maybe earlier than that.
But I think it's wrong. It runs contrary to the plain sense of our prayers to the saints, and to what I take to be the normal sense of the faithful. It requires us to believe that something that is not a person and not me could have my thoughts and desires, and could receive the reward or punishment that I deserve.
So I hold a different view, "survivalism", the view that I will survive my death as my soul. But for that view to be true, I've argued in the paper I linked above, human persons only need their souls to exist. On this view, then, my soul is naturally meant to inform and express itself in a body. But it doesn't strictly speaking need the body to exist and to be a person. My soul needs the body to implement most of its powers--powers like breathing and digestion and sensation and so forth--but it doesn't need the body to be a human person. My soul (which is me) is meant to inform matter, so it forms one substance, one thing, with matter--this isn't a version of substance dualism, the view that I am made of two complete things, soul and body. For this reason, I can say that I am my body. I'll be radically incomplete without the body, so the Resurrection is still deeply important. But I'll still be me without the body. For this reason, I can say that I have my body.
On this view, then, what I am is rightly called an angel--that is, a being that is a purely immaterial substance, an intelligence--albeit one that is naturally capable of having a body, unlike all the higher angels. But if I'm an angel now (albeit an embodied one and one that is also an animal--that is, a sensing, naturally material thing), then there's no problem with calling me an angel when I'm dead.
Some of you might worry that this does sound too much like dualism--a view that, in the form given by the 17th century philosopher Rene Descartes, has been blamed for many modern woes. Some Catholics say that when Descartes argued that the soul is an entirely distinct thing from the body, it led to the view that the body is a thing that we can entirely manipulate and use however we like, with no natural law inherent in it, and with no meaning other than what we imbue it with. (I don't think laying the blame for this widespread modern error at Descartes' feet is at all fair, but that's an issue for another day.) But the problem here is not the view that I am my soul--rather, it's a particular problem with how matter is conceived (as valueless and as entirely describable in purely mathematical terms and as raw stuff not ordered to any particular ends or purposes aside from the purposes we decide on), and with how the connection between soul and body is conceived (with the soul manipulating the body as something entirely exterior to itself, rather than forming for itself a body such that that body has natural ends that must be followed). My view explicitly denies these claims, so it doesn't leave us open to any of these modern errors.