Yes, laptops can have a thermal cut-out (or at least my old Athlon XP system does). Usually this will manifest itself as the laptop suddenly turning off without warning, as opposed to refusing to power on altogether (you don't say if it refuses to power on from "cold" i.e. not being used for a while, or not).
Even if it has a thermal cut-out, you should still see power lights on the machine (a/c power connected) - it's just that it will refuse to turn on.
If it's not thermal, then the most likely would be either the a/c power supply or the battery. You'll need to try combinations of running the machine just on battery, just on a/c (take the battery out completely), with battery charging etc. You should be able to narrow it down and replace the bit that's broken - ebay would be the best bet, especially for a power adapter.
When my laptop's thermal protection steps in it's pretty obvious. Feel under the laptop roughly where the cpu is - in the middle towards the back - if it's a thermal issue it will be really hot to the touch. You need to promote airflow - you can try clearing the vents/grills ( a blast of compressed air or the attentions of your vacuum nozzle will do the trick). Otherwise, if you're using it on your lap, try to place it on something flat, like a tray, rather than on a cushion / pillow which may be blocking the vents. If it does cut out, you just have to wait for it to cool - you can try closing the lid and leaving it upside down which may let some heat dissipate faster.
You've said that it's infact refusing to power on from cold, which means it's probably not thermal (unless your ambient temperature is > 120F). If you have no battery, just A/C, you should still see the A/C light even when the machine is off. No lights imply a dodgy power supply.
Make sure the cables on the supply (plug to transformer, transformer to laptop connector) are good with no obvious signs of damage, especially around the ends of the cable (where it joins the plug, transformer, connector) - this can flex and break over time, especially if you wind it up a lot and stuff it in a bag. Also check the power socket on the laptop - make sure the central pin isn't loose - again these can fail over time.
Ebay would be the best source of a power supply, especially for an older machine - you should be able to get a generic replacement: you need to check the output of the transformer and that the connector is compatible (good sellers will list what machines it fits). If you find the pin in the power socket on the laptop itself is broken, then this is much harder to fix - you can source them from radio shack (us) / maplins (uk) but it means complete disassembly and soldering on of the new fitting - not easy.