Hassel,
I recently bought a 2nd-hand laptop (IBM 760 XL) and it randomly froze up, sometimes during booting, sometimes when running a DOS-based program, sometimes when running a Windows-based program, sometimes when on the 'net. Randomly froze, or fatal exception errors. Erratic.
I had put in a second RAM chip to fatten it up to 64Mb. Originally it had 32Mb only but we found a 48Mb chip that took it to the full 64Mb which the BIOS could recognise, it ignored the additional 12Mb.
With one RAM chip in (either one, it didn't matter) the machine performed faultlessly. But put both in and the machine became the most flaky, frustrating piece of hardware I have ever seen! We swapped the chips around in the slots, no difference.
Finally I just removed the 32Mb chip and run with 48Mb RAM on one chip now...no more problems, runs like a dream!
Suggest try swapping RAM chips around in the slots if you can access. It might just be the problem. Certainly was in my case. Also, non-matching RAM chips may be the problem. RAM troubles of some sort sound like the underlying cause of your problems.
Hope this helps.