Allegedly, cheap filters can degrade the image if you're shooting long lenses; the longer the lens, the more the degradation. I've not tested this formally, so I don't know, but I have tested resolution with 'normal' lenses, and I needed to use 3mm window glass to see a difference. Everything else (glass or plastic) was fine. Also, anecdotally, I rarely shoot my 200/3 Series 1 without a (Soviet-era) orange filter, almost always with the camera (Nikon F) on a tripod, and it looks pretty sharp to me. The only time I remove UV filters is when the sun or another light source is in shot -- and then, not always. Certainly, reflections are far more important than loss of sharpness, and even reflection problems are rarely important. Often they can arise without the filter, in any case.
Ctein, who is a far better experimentalist than I, agrees about general image degradation. But, as ever, few people ever do conduct formal experiments, and of those that do, you can sometimes see from their reported methodology that the experiment was worthless. Usually, those who worry about image degradation rely on one of three things: anecdotal evidence, or what 'everyone knows', or theory.
In the last case, yes, there must always theoretically be some image degradation, but everyone who has set up a reliable experiment has come to the same conclusions as Ctein and me, viz., that you can't actually demonstrate this theoretical loss in a real world image, so the theory is not worth worrying about.
Like Pickett, I've smashed filters (well, I've smashed one on the 200/3 and Frances smashed the other on a 35/2,8 PC Nikkor) and in both cases the camera lens was unhurt.
Cheers,
R.