GPS used a 10 bit field to encode the week number in each GPS time message, which means that a maximum of 1,024 weeks (19.7 years), could be handled. Each of these periods is known in GPS terms as an "epoch".
The first GPS satellites went live on 6 January 1980, meaning that the first epoch of GPS time lasted until 21 August 1999. We are now nearing the end of the second epoch, which will fall on the 6 April 2019. That means from that date onwards, we are likely to start seeing rollover problems in GPS receivers that aren't programmed to cope with the week number reset.
One of the things that makes this issue different from the Millennium Bug is that the impact won't necessarily be felt on rollover day itself. In fact, it's much more likely that an affected receiver won't start outputting erroneous data until long after the 6 April 2019.
That's because many receiver manufacturers have sought to maximise the default lifespan of their receivers by implementing the 1,024-week limit from the date the firmware was compiled, rather than from the date the current GPS epoch began.
In effect this means that older GPS receivers will operate normally for almost 20 years before problems begin to occur -- and if firmware is implemented in this manner, no issue is likely to be seen when the GPS epoch changes.
Correction, 13 bits should be enough for anybody:
The GPS modernization program is an ongoing, multibillion-dollar effort to upgrade the features and overall performance of the Global Positioning System. The upgraded features include new civilian and military GPS signals. To improve the situation regarding Week Number Roll Over, message types (CNAV and MNAV) use a 13-bit field to represent the GPS week number and newer GPS receivers that utilise that 13-bit field will not have a problem with 1,024-week epochs.
Update: A couple dozen flights were grounded because of this.