Bad capacitors are extremely common in budget brand or old TVs, but this is not a common symptom of that. Generally once they start going bad, you might get funny behavior for a short while but then the TV completely stops working, not ever coming back not rather than just taking a long time.
What is more likely depends on whether it's being turned on by the remote or a button on the set. On the set the button may be misaligned either from the factory or worsened through the stress of pressing it hard (or poor design subjects it to this) or in the case of the remote, a buildup of tobacco tar on either end, remote or receiver panel, or time of day (sunlight in the room), low remote batteries, etc.
Then again for all we know the problem isn't even the TV. It could be that the source, a CATV tuner or internet connected box, etc, is not on or not receiving signal. Never underestimate people's ability to overlook something the more complex the setup is.