Brake fluid could be low. Check for leaky brake lines under the car also. Quite often older cars have rusty brake lines that leak fluid sometimes just a little bit but these should be repaired as they can blow apart when you hit the brakes hard and cause an accident. Check your emergency brake and make sure it as pulled all the way off also.
See sources and related links below to lookup both vehicles and compare bulb numbers.
If oil level is ok - could be a bad sensor
How do you replace the third rear brake light in a 2005 buick allure
The 2001 Buick Century brake light switch is located beneath the dashboard, on the drivers side. The brake light switch should be labeled as such.
I'd check the sensor for the emergency brake. It's probably faulty.
I have 05 with no brake lights
VERY CAREFULLY
I think the light assembly screws out from the inside of the trunk.
A short in the Brake light circuit? Bad brake light switch?
The correct size is 3057.
Cut the wire
The button on your dashboard