NavList:

The only GPS time correction I am aware of was explained to me by a Lowrance or Garmin tech (I forget).  I do not know what or how many GPS recievers work exactly this way.

They explained to me that when powered down, the GPS reverts to its own internal clock, that runs on its own memory battery, similar to a computer motherboard.  My lowrance has external dash kill switches on the antenna and processor, so it may well have been this one.

Basically it does its best with a high degree of accuracy...like a digital watch...when it is powered down.

Upon boot, it comes up immediately with its own stored time, displayed as UTC.  As the GPS acquires satellites and gets itself updated, it downloads the most currently corrected time.  At that moment you will see, if the internal clock is wrong a change in time. That was where the magic fourteen minutes I was given came from. They told me it should not take longer than that to update the subroutine.

Does it happen...yes.  I watched it and tracked it with my digital casio.  Last winter the weather was so bad up here in new england the boat was out of action in Noank harbor from week 3 january to late march, due to ice, cold, and a touch of despair.

The GPS was off.  So when I turned it on to do celestial sights for the heck on the first warmer march days..and sat and watched it, my casio propped up beside.  Sure enough, the atomic g-shock and GPS were out of sync 3 seconds when started up...then between ten and fifteen minutes came a clear cut "jump" of two seconds, that brought them back into their usual one second discrepency.

This appears to confirm what I was told. The internal clock may have a slight timing error, corrected when it updates.

Now with all that said....it seems utterly useless as an explanation for your question.  As the the prolonged time for a trackable error seems missing from your equation.

The GPS of course tracks GPS time and supplies UTC for convenience.  GPS was started on a corrected UTC time, but tracks TAI (Time Atomic International)  UTC and TAI are about thirty seconds apart..... and GPS is somewhere in the middle. Accumulating an ever increasing gap. At to that that UTC and UT1(equivelent to GMT) wobble around each other + or - corrected by leap seconds (accounted for by the DUT1 correction from the naval observatory ) ....and the whole thing is a complete profusion of confusion. 

Interestingly, I have heard of a rare GPS running amok with totally incorrect location (heresay, not observed by myself). This does support, if true, the case for celestial (with a seperate timepiece) as a cross check.  If you blindly trust one GPS and it is wrong....in open sea well...yikes!

Thinking about fussing with excessive accuracy....I am OCD about it....LOL....must be the engineer in me. I can't seem to control my urges to correct time to the gnat's eyeball, even when it plays zero practical significance.  I guess I am testing myself....trying to prove that like the polynesians I could somehow safely arrive at the equivelant of a large anthill in the middle of the pacific ocean.

Yes I know...I am not in the pacific...not lookng for an anthill....and but what if I was??? LOL... its all about being the best you can be...I guess...