Nearly all tv faults are "dry joints" which means a soldered joint on one of the components has become dry and doesnt make a good connection, this can often be checked by banging the set to see if it upsets it even more, but that doesnt always work.
this definatley sound like a dry joint when it gets hot it starts to disconnect, if this is the case you need a soldering iron and an eye for what to look for, and this is where the problem is, unless you know what a dry joint looks like you probably wont spot it. but if you find it, it taks 2 seconds to fix, you just have to reflow some solder over it.
The most common place for these to happen is around the power supply area and the high voltage area as these generate the most heat and suffer from thermal movement.
All it needs is taking the back off and looking at the underside of the circuit board 'normally green with silver spots', especially around anywhere that looks slightly darker i.e hot areas.
Please DO NOT do anything with out unplugging the tv first and leaving for 20 minutes to allow everything inside to become safe to work on.
I fixed my 14 year old tv with a dry joint years ago, still going strong (not that i get the benefit of wide screen or surround sound, but im to tight buy new when the old works fine)
gadget31, July 2006