Laptop does not wake up

I have been having trouble with waking up my laptop when it is only on Standby (in other words when I just shut the lid).

I use Windows XP but a lot of people are reporting similar problems with Vista as well.

My experiments show that this is most likely to happen when the laptop is put on Standby while it is connected to the power supply (ie, the mains).

99% of the time I have Firefox open.

Recently, I've figured out that Firefox is the root cause somehow for this. I did exactly the same things that would cause the problem but without the Firefox running, and so far the results are encouraging.