What could cause my light and TV to turn off, then back on?
You might be at risk of a house fire, and I believe you should call an electrician to come out and diagnose the problem, unless you have the diagnostic skills yourself. Maybe it's something that would go on for years without causing a fire. But maybe not, and either way it's annoying or (much) worse.

You could have a broken wire inside a wall, or in a junction box. This can happen if NM cable is pulled over-enthusiastically when there's a kink in it, especially if it's #14 wire.  Or it could be a bad connection in a junction box. You could even have a bad connection inside your service panel, or a bad breaker. I assume both the TV and the light are on the same circuit, since they seem to be the only things affected? 

Electrical connections should be tight, no wiggling, with plenty of pressure and plenty of metal in contact.

A bad connection would cause heat (potentially a lot of it). The heat would cause thermal expansion, which might be breaking the circuit by physically moving bits of metal apart. Then as it cools back down, the parts separated by the thermal expansion touch again and the light and TV come back on. But the connection is bad, more heat, thermal expansion, repeat. If this is what's going on, you are potentially at risk of a fire, and you should call an electrician.

Another kind of weird (very unlikely) possibility is that the TV is somehow wired through the light fixtures. Does the TV come on at all if the light switch is off, or does the light have to be on for the TV to work at all? I only ask because you explicitly mentioned turning the light on before turning the TV on. Some light fixtures are designed with thermal protection and shut off if they overheat. If the light is shutting off because it got too hot, and the TV is wired through the light fixture somehow (serially? Or using a neutral that gets switched off?), the TV would go off and come back on with the light. As I said, though, this is unlikely and you almost definitely have a bad connection somewhere, which is a fire in the making.

Regardless, I really think it's time to call an electrician. If it's something that wouldn't cause a fire, you get it fixed and you're out a few bucks. If it's a fire hazard, then you will have potentially saved lives.