If it's a large heatsink, it should save it long enough for you to stop what you're doing and safely shut down. If it's just the stock heatsink (since the fins are so close together), I wouldn't expect it to last as long as with a larger, more open design which can radiate the heat better; the stock heatsink depends on the airflow, but it should still be enough to get the system shut down. It's part of the expected design of any good heatsink that it should at least provide "safe" operation, though not optimal. Athlons can safely reach a pretty high temperature without permanent damage.
When I set up a system using the stock thermal pad, I boot it up with the fan unplugged before I close everything up, and let it sit in the CMOS setup monitoring screen and watch the temperature rise, to allow the thermal pad to reach the phase change temperature better than it would in actual use. The CMOS setup runs the CPU at a medium to high usage, so it boosts the temperature better than running the OS idle.
You'd be pushing it trying to run the system at 100% for more than a short time after you hear the fan die. It might not do immediate severe damage, but it may cause damage that results in minor, hard to track down instabilities later.