Bad thing is that for higher-order ones, you have one big cell with polynomials of degree 3 or 4 in it. Bad because if you have shadows that fall across the cell, you have a discontinuity, so you'll get Gibbs' phenomenon and will ring at shadow boundaries. Of course, you can smooth it out with a filter. Expensive representations for the cornell box are okay. For a tree with a zillion polys and 4th degree poly on each, that's big.
Small surfaces are tough.
If you go to higher-order or different types of wavelets, it gets more expensive, and you don't have good edge continuity. Not obvious that higher-order or differnet bases are a win at all. Just do PW linear for ass't. Can do higher-order or weird ones if you're eager to do so. Now let's talk about meshes.