If it helps I had a similar problem a couple years ago. For clarification...
1. Turn PC on, fans spin, lights come on, doesn't post.
2. Reboot a few times, ends up posting, works fine.
Later on it developed symptom #3.
3. #2 doesn't work, unplug PSU, hold power switch for about 30s or wait 5-10 mins, replug it up and retry, boots fine.
It did this intermittently for about 6 months. Eventually it shut off on it's own. Upon trying to reboot fans wouldn't spin up, lights wouldn't turn on and the system wouldn't even attempt to power on. #3 above provided zero results. In the end swapping in another PSU fixed the problem. This lead me to believe the 6 month period was due to some component in the PSU starting to fail. Eventually it went from it's last leg to dead.
Basically, I'd rule out the PSU. I won't say it definitely is your problem because these type of issues can be tricky to troubleshoot. I do find the PSU is often a good option to rule out though. They're relatively cheap compared to a lot of components, at least to swap out, and they don't tend to have a lot in the way of compatibility concerns. Not like various other components anyway (chips, boards, ram, etc.).
The other option would be to ensure it's not the board. Faulty boards can lead to some very interesting behavior.
Before swapping out components removing all non-essential components and trying to boot/check behavior is a good starting point. You should be able to rule out most stuff by taking this step. Unfortunately, this isn't often helpful when a system is behaving normally sometimes but not others. Ruling out other obvious suspects like over-heating, doing a visual inspection of components and things of this nature goes without saying.