Yes, it may be possible, depending on the rim size. Most older MTBs had "1.5 inch" rims, which have a <1.5" inner diameter. Look at Sheldon Brown's site for guidance: http://www.sheldonbrown.com/tire-sizing.html .

Don't overlook 1.5", I had done the same exact thing recently for my wife, and I was surprised how fast a 1.5" semi-slick tire will roll. The ride is definitely dominated by the geometry/fit of the bike, as it is more upright than my road bike. The rolling resistance doesn't really play into it too much for a relatively slick tire with higher pressure. She has no trouble getting up to 17-19 mph, but above that becomes tough because of the wind resistance.

There is also no reason why a 1.5" tire should be expensive. I have purchased both a set of these: http://www.amazon.com/gp/product/B00...ef=oss_product and a similar Kenda set in a LBS for $40. Both are very fast, and the limiting factor on the bikes are the positioning and gearing not the tires.