What happens is the capacitors (in general) are very sensitive to temperature, lets say we have a capacitor that was designed to work in a range of 10ºC up to 80ºC
After lot of time of use, when is weared out that range could be something like 27ºC up to 43ºC
If you are in a cold day in winter, lets say with ambient temperature of 24ºC then inmediatly when you try to boot the device... is not going to work (because ambient temperature is lower than the range needed by the capacitor)
You press the button and the device tryes to boot... and inmediatly it turns off... and usually tryes to boot again... and inmediatly turns off again... and so on in a loop... and while doing that loop the circuit board (and the capacitor itself) starts to heat up a bit... so after few minutes doing that loop the device turns on
This happened to me in few PC monitors, with one of them i was turning it on at the morning and it started doing an infinite loop trying to turn on
I was lazy (and i imagined which problem it had) so i was using the monitor like that for several days
The first days it was booting in 3 minutes or so... and after a week it was taking 10 minutes to boot (lol, at this point i had to fix it and stop being a lazy bastard... i opened it, changed the capacitors, and fixed it)
Im not saying that this happens always, but is very usual... and actually sometimes it happens completly the opposite, the capacitor only works when the device is cold... but when it heats up it starts failing
This happened to me with PC power supplies, at the first 1 or 2 hours of use the PC was working fine, but after that the hdds was doing the typical "click" noises that means the hdd needle went to the "parking" position and the hdd have been disconnected
Yeah, this was causing the classic "blue screens of death" in windows and eventually caused me some damaged "bad sectors" in the hdds
Here the problem is the opposite.. the capacitor cant operate over lets say... 45ºC
Moral of the story... if the temperature goes out of the range of the capacitor (either for excess or for being not enought) the power line becomes unestable
You never know if is one or the other, but there are some cases, like what
@Revak3115 is saying, his PS3 only works fine when is hot (and probably in summer better than winter), it looks a typical example of what im saying