A TV that runs fine, then shuts off after ten or thirty minutes, then plays again after it cools is describing a thermal fault. The timing is the diagnosis.
Two shapes are common. The first is a failing capacitor, usually in the power supply, that drifts out of spec as it warms and drops a rail. The set behaves perfectly cold and dies predictably hot. The second is a thermal protection trip: the main board or the backlight driver overheating because of a weak thermal pad, a dead fan on the big sets, or a dust-choked vent path.
What you can do before opening anything: time the shutdowns. How many minutes cold, how many after a restart when the set is already warm. Short intervals on warm restarts point at heat; identical intervals every single time point more at a timer or protection latch on the main board.
Then, with the set unplugged and discharged, look at the power board for domed caps, brown residue, or darkened board areas. Replacing capacitors in a TV power board is a real and satisfying repair at component cost, though the working voltage and heat ratings have to match or exceed the originals.
Post your set's model and the timing pattern; that usually tells people whether they are looking at a two dollar part or a new main board.