Always make sure we define the problem accurately. I'm assuming that we're all talking about electrical 'power', and not that the computer is unable to support the card because of a lowered processing power.
A bad connection causing a significant power loss would be all too evident. To lose 'power' across an unwanted resistance would almost always cause fluctuating voltages on the device being supplied, with associated 'electrical noise'. This could be soaked up in the video card's smoothing, but I have my doubts.
However, years on the electronics workbench just makes one expect the unexpected. So, cleaning/re-seating contacts is a first and certainly cheap step.
For the PSU output - on one or more lines - to have dropped just a small, but stable amount, seems a little unlikely, but not impossible.
If it were a voltage problem, my gut feeling would be that the Nvidia sensing might always be right on the cusp of giving a warning, and has just gone into the 'react' bracket.
There is a Chinese Digital Volt Meter (with other usual functions plus a few) - being sold round the world now - $1.49 here! - and yes, that's not a mistake. It even comes with a free 9v battery. It agrees with my Avo and my Fluke almost exactly. Taking readings of the voltages shouldn't be too hard, even for someone with little experience. Voltages are always a good start to fault-finding.
Turning the unit to AC, and measuring again, should show zero in a perfect world. Any 'noise' might show up as an AC voltage. This would be a significant warning flag, though as I've implied, a 'rough' input supply to the card would probably have caused more chaos than just a voltage warning.