Sub placement inside the room and poor subwoofer design are the two biggest sins. I don't know which is worse.... wait, yes I do: a poorly designed and built sub can never be made to sound any better.
@Pogre is spot on in his assessment in terms of the overall quality a better sub can deliver. I barely use mine in that they are set around 1/4 - 1/3 gain... yeas I can rattle the house, but there is no real reason to do that. At the level I use my subs, there is no distortion to speak of and they sound amazing.
To your question about a new receiver, this is tough. I don't think you will experience what I did shifting from a >10yr old Onkyo HTIB receiver to a new Marantz. I noticed this detail in particular as I was still using the same speakers and sub from that HTIB: the Marantz processing was significantly cleaner and the old Speakers sounded better and the Sub played louder with a less distorted signal. We are talking in my case of changing from an outdated 10+yr old $250 AVR to a modern mid-high level AVR retailing at $1500. That is pretty significant.