YES. Bad filters can have all sorts of effects. It is after all a glass/laminate and if the medium is defective it will affect the image. Normal UV filters from Hoya cost around $10-20, so there is no use going for cheaper filters. Polarizing filters cost a bit more, but they are still below $75 for most sizes. As long as the filters are of good quality and kept clean they have minimal impact on the image.Stupid question...
Could some cheap filter make my photos soft, and unsharp?
Now there is an interesting class of filters called "Softars". They were used in the film era to soften up portraits, but are rarely used in digital age, as software can do that for you (and more).