Depends on the genre, but in general a low budget doesn't preclude high quality. With the festival system for indie films, a number of ultra low budget movies have become big hits. Napoleon Dynamite's a great example, so are films like the Irish "Once" which won an oscar and was made for practically nothing.
depends on what your definition of a big budget film is? i can list a lot of indie movies that were great that didn't cost that much to make. there's "district 9", "kick-ass", "american splendor", "history of violence", "pulp fiction", "cyrus", "blair witch project" and many others. just to name to a few.
It seems a butt-load of that budget is going right into advertising, so unfortuantley alot of low-budget films can't compare, which is sad, considering low-budget films are some of the most abstract/lustrous/beautiful tales told. The hard part is finding them (and sometimes understanding them) .
Budget shouldn't matter if the storyline is good to start with.
A prime example of a good movie made on a very small budget is the original "The Terminator". James Cameron made that movie on around $6,500,000 which is a fraction of the $58,000,000 budget that "The Cotton Club" had from the same year.