I've had the same problem happen in the past. From my experience, it seems to take a guy who isn't on the depth chart and put him in at the vacant position, there might be something to the sim engine deciding not to disrupt the rest of the lineup to fill the necessary position. Either way, I've started putting a third-stringer in over the past couple of seasons rather than starter-backup-starter. He'll really only come in during garbage time, and occasionally if foul trouble makes the two players ahead of him unavailable, but all of that is easily overlooked when a situation like the one you just had happens.
If you're interested, look at my player Kris Allen's minutes over the last two seasons. I have him as my "reserve" at PF and C every game and I don't think he's played a high-leverage minute yet where my better interior players have been available.