|Thanks for the responses.|
I have replaced the hard drive in it with a new 80GB 7200RPM, since the original drive crapped out. It does not have built in wifi, but a PCMCIA wifi card. This problem occurred before the hard drive was replaced, and continued afterwords.
I ran some memory and CPU tests using Ultimate Boot CD, and they all came back fine on the first iterations of each test.
The cooling fan comes on a few time an hour. When it does come on, the air blowing out will be extremely hot, and that's even sitting on a flat surface with plenty of ventilation.
I read somewhere that some of the Thinkpad laptops had faulty motherboards that failed when both memory slots were used, but the T23 was not on that list.
I will try to use the Puppy Linux and see if it finds anything.