During my son's recruiting process, a couple of schools raised their offers as they became aware of more schools entering the mix. He never told any of the RC's about other offers, but they knew. A few of them also made the comment that if the offer was the final determinant in his decision, and it was within a few thousand $'s, to let them know so they could see if there was more could do.
It was never a "bidding" scenario, but if you think about it, so to some degree, any kid with more than one offer is in a bit of a bidding war of sorts. In my son's case, the situation and offer he accepted were both excellent and not much could have been done to improve either, so he never did "seek" additional offers from any other schools.
I'm sure the school in your scenario is indicating something similar in that if you get another offer that you're considering, and the primary driver is the size of the offer, they'd like a chance to match it, beat it or pass if it's more than they're willing to go.