There are two typical kinds of symptoms: power on click but nothing else happens or a tick-tick-tick sound indicating cycling of the low voltage (line regulator) but lack of startup horizontal drive.
Check the voltage on the horizontal output transistor (HOT). If no voltage is present, there may be a blown fuse or open fusable resistor - and probably a shorted HOT.
However, if the voltage is normal (or high) - usually 100-150 V, then there is likely a problem with the startup circuit not providing initial base drive to the HOT.
The startup circuits may take several forms:
The startup circuit may operate off of the standby power supply or voltage derived from non-isolated input. Be careful - of course, use an isolation transformer whenever working on TVs and especially for power supply problems.
Note that one common way of verifying that this is a startup problem is to inject a 15 kHz signal directly into the HOT base or driver circuit (just for a second or two). If the TV then starts up and continues to run, you know that it is a startup problem. However, this is a risky procedure for both you and the TV. See the section: Bypassing the Startup Circuit.
I have also seen startup circuits that were designed to fail. Turning the TV on and off multiple times would exceed the power ratings of the components in the startup circuit. Some Zenith models have this 'feature'.
When this situation exists, it could be that the circuit is not providing the proper drive or that due to some other circuit condition, the drive is not always sufficient to get the secondary supplies going to the point that the normal circuits take over.
I would still check for bad connections - prod the circuit board with an insulated stick when the problem reoccurs.