It depends on the lender and other factors. But it's often very difficult to borrow less than $50,000.
If you're looking at a property where the amount to borrow would be less than lenders would be willing to lend, there are a number of other ways to handle it. The easiest would be owner financing. The seller becomes the "bank" and you'd pay him/her back directly. You might also be able to find private financing. That could be an individual with some IRA money, for instance, or a private lender. The interest rate is likely to be higher than you could find on the open market (for higher-amount loans), but the difference between monthly payments on a $40,000 loan at 4.5% versus, say, 6%, is insignificant.
Hope that helps.