There might be something in the house equipment that you're not aware of..
I wouldn't suspect that the cable box is tripping the GFCI. If it was the cable box it would almost certainly trip the GFCI as soon as you plugged it into the GFCI.
I already started typing the following up earlier. If your interested, it will explain what the GFCI does and walk you through process of narrowing down the source of the problem.
I don't know how much experience you ( or others ) have with wiring or GFCI outlets so here is a quick overview.
1) Circuit breakers are designed to protect the building from fire, over current / over heating. They only measure the demand on the hot wire.
2) GFCIs protect people from electrocution. By measuring the power on both the hot and neutral wires a GFCI can react to smaller variances in power and also respond more quickly. Generally, if there is a variation of 5 milliamps or more between the hot and neutral wires the GFCI trips. The advantage is that if you happen to be grounded and touch a hot wire or hot enclosure etc but only draw, for example, 18 amps on a 20 amp circuit the GFCI would trip but the circuit breaker wouldn't because the circuit hasn't gone above a demand of 20 amps.
3) Usually, but not always, the electrician will run a circuit from the main panel box directly to the GFCI. The wires from that circuit connect to the " line side" terminals on the GFCI. Anything that plugs directly to that GFCI will be protected.
Also on the GFCI are two more terminals "the load side" . If the electrician used that same GFCI to protect additional standard type duplex outlets or equipment (very likely because its cheaper) something further down the circuit could be tripping the GFCI. Maybe a automatic sprinkler, central air humidifier, pump at a fish pond, hot water circulator pump etc..
With that in mind, if you want to try isolating the problem, turn the power off at the panel box, remove the two screws that mount the GFCI to the outlet box. Do you see three or five wires connected to the five terminal screws on the GFCI? If you see five, I would disconnect the two on the load side ( its labeled ), reinstall the GFCI in the box and plug everything back in like it was when it tripped. If it doesn't trip over the next couple of days then something else further down the line ( load side) is causing the problem. If it does trip, either the GFCI is faulty ( unlikely because of what you described above ) or whatever is plugged into that GFCI is letting power trickle to ground instead of the neutral.
If you see only three wires on the GFCI when you initially take a look at the GFCI then the problem is a defective GFCI or one of the items that your plugging directly in to the GFCI.
BTW, if the GFCI did have five wires on it and you removed the two on the load side you killed the rest of the outlets on that circuit. Make sure that we didn't take out a refrig, dialysis machine or something else that's important.