Foreclosures can take 3 1/2 to 4 months to complete, from the time the Notice of Default is filed. During that time, the bank can collect rent and, as such, give you a notice to vacate if it so desires. If you refuse to move, the bank can formally serve you with an eviction notice, haul you into court, damage your credit report and send the sheriff out to physically remove you. Sometimes, to avoid that hassle and expense, a bank will figure it will spend less money to offer to help you move. But that's not a route most people want to go.
If your home is in foreclosure, you can call a title company and ask its customer service department to pull up the Notice of Default. Title companies are happy to help the public on a one-by-one basis. Ask for the date of recording. You can probably work out an arrangement to stay in the property for about three months after that date.
I can only speak from my personal expierence and I give cash for keys to all people that are living in homes in order to get them out as soon as possible, if the tenant however, becomes difficult in terms of moving out, we just start the eviction process. I would not use it as a profit center, but a lot of banks, World Savings/Wachovia, Countrywide, etc. will offer money so you don't wreck the house.