I know a student who got into his first-choice program this way. They were very interested in him based on his work experience and transcripts, but after taking the GMAT twice he still couldn't get it high enough to meet their minimum. Since admitting him based on that GMAT score would have lowered their average score and might have therefore hurt the program's ranking (it's a very small program, so one score would make a difference) they asked him to take the GRE instead and admitted him based on that score. I'm not sure if this is ALWAYS the case, but for this guy, the school was definitely making an exception to their published policy in order to circumvent the statistical impact of admitting a student with a low GMAT score.

If this is something you're interested in doing, I'd recommend asking the admissions office directly; it seems to me that programs may do this more on a case-by-case basis.