It depends on the vote. Some GPS will have problems like the date NOT rolling over at midnight exactly but a few samples, so if there is no network or something else, bad things can happen. If there is a sanity check (and assumes thorazine, straitjackets and padded cells), maybe, but the only thing you should use a GPS for is the PPS signal itself. The "fix" will usually occur well after (sometimes a half second).
There is the accuracy v.s. precision dichotomy. The PPS can make the time value very precise (assuming it doesn't emit it when the fix isn't good). But the NMEA nonsense can result in inaccuracies - of a day as in the case of the above (a Garmin 5hz unit I used).
A Dallas Semiconductor I2C RTC chip
can be tuned and synced so its "PPS" syncs with the GPS and it would be far better a time source (2PPM!) than any raw GPS not designed for time (SkyTraq has a 10nS GPS chip designed for just this)