The accuracy of GPS isn't really a disadvantage, except for people in the military who are paranoid by profession and spend lots of time imagining ways in which their own technologies could be used against them. The military philosophy is to keep every technology classified forever so that the bad guys can never use it (and if civilians must suffer from not having it, so be it).
Fortunately, cooler heads prevailed with GPS, which is why the military was ordered to make it available to the civilian world.
The overreliance on GPS is a concern, and there is no obvious back-up. Galileo will not be a back-up, since it's just another version of the same technology. The problem is that there may be no back-up for satellite navigation in general, especially as government authorities foolishly decide to retire other forms of navigation to save money. If you have VORs or NDBs, you can still find your way home in bad weather. If you have only GPS, and GPS stops working, you're lost.
There are some circumstances in which GPS accuracy can lead to problems. One problem, already adumbrated here, is that GPS normally is used in conjunction with a database, and the overall accuracy of GPS in practice depends not only on the satellite longitude and latitude, but also on the accuracy of the database that tells equipment what lies at that latitude and longitude. This is why GPS might put you a mile north of an airport: chances are that the GPS was right on the mark, but the database was substantially in error.
Another drawback to accuracy is that it can put two aircraft in exactly the same place at the same time, which is bad. Two aircraft flying towards Big Mountain might still be thousands of feet from each other, but two aircraft flying towards the BGRSK waypoint might well blend together when they reach it, if they happen to coincide in altitude and time.
Still another drawback is excessive reliance on and trust in the technology by pilots who don't really understand how GPS works and what the potential problems with it might be. They come to believe that GPS is always perfectly accurate, and they might find out too late that it isn't. Some pilots talk as though they trust GPS altitude more than altimeter altitude, for example, when in fact the altimeter is orders of magnitude more accurate than GPS for altitude.
There's also too much willingness to trust GPS for approaches (in theory or in practice), as a substitute for ILS. But ordinary GPS cannot come close to ILS for accuracy, especially for precision approaches. Using GPS in place of ILS requires LAAS (which is not GPS at all, but merely a supplementary technology), and even that is delicate. And no, things like WAAS or RAIM do not make GPS anywhere close to foolproof.
As always, the smart aviator will use every available tool to maintain maximum safety in the air. GPS is all well and good, but it's still extremely important to have a Plan B. Airliners have this built in to some extent, with flight management systems that can combine information from multiple sources (GPS, IRS, VORs) to arrive at highly reliable conclusions; but private pilots in small aircraft who have got into the habit of navigating solely by GPS can get into deep trouble (as multiple NTSB accident reports demonstrate).