It is in principle possible that two different gravitational waves from independent sources could arrive simultaneously at the LIGO detectors, which would then probably pose problems in identifying the precise origin of the signal.
However, the probability of this is vanishingly small, because LIGO consists of two detectors, not just one, and the signal would have to arrive simultaneously at both detectors for the problem to arise. This places some very tight constraints on the geometry (basically, given one source's direction, the other source is constrained to a cone around the inter-detector axis) along with the timing of the signal.
Moreover, given that the current pace of detection has the detectable signals spaced from each other by weeks or months and lasting fractions of a second, a signal coincidence is has a vanishing likelihood without a change in technology. And, as detectors improve and the sensibility goes up to thresholds where simultaneous detections start becoming more likely, we will also have more detectors placed in different locations, which places much more stringent requirements on the spatial coincidence, in essence asking more and more for strictly collinear sources. (And also: this issue isn't unique to gravitational-wave astronomy. Optical astronomy has this problem all the time, and it's often quite tricky to solve.)
Nevertheless, the question of what would happen if two signals were to overlap is indeed interesting. For many of the existing signals, each merger's chirp occupies a distinctive pattern in both time and frequency, which means that there's a strong chance that we'd be able to separate them out spectrally, i.e. by which frequency bands each signal occupies. There are limits to how far you can push this if the signals partially coincide, but as detector sensibility improves so does the availability of raw data with which to do finely-tuned data analysis.