Can two earthquakes happen at the same time?
The 2016 paper "A case for historic joint rupture of the San Andreas and San Jacinto faults" reports of research suggesting that simultaneous large earthquakes occurred in 1812 in California, the researchers used


  dynamic rupture modeling to show that paleoseismic and historic observations associated with the earthquake of 8 December 1812 can be explained by joint rupture of the SAF and SJF.


(SAF = San Andreas Fault and SJF = San Jacinto Fault)

But the author stresses that what they have is a model, rather than a certainty.

Alot of research centres around the notion of 'near-simultaneous earthquakes', such as the 2009 South Pacific earthquake, as described in the article "Double Shake: Multiple, Nearly Simultaneous Earthquakes Triggered Deadly 2009 Tsunami" where studies suggest that at least 2 quakes of similar magnitude occurred within 50km of eachother, separated by minutes - another similar study suggests that one of the quakes 'obscured' another nearby rupture of equal or higher magnitude.

Italy is another place where simultaneous or near-simultaneous earthquakes seem to occur and resulted in several studies, one of which being reported in the article "Likely near-simultaneous earthquakes complicate seismic hazard planning for Italy", stating that


  It's very important to consider this scenario of earthquakes, occurring possibly seconds apart, one immediately after another


Stating that this scenario would have an 'amplification' effect of the the effects of the shhaking (hence, the researchers' concern).  A particular example they cite is


  The 1980 Irpinia earthquakes included a sequence of three events, occurring at intervals within 20 seconds of each other.


Another compelling example, reported in the article "Sumatran strike-slip earthquakes challenge seismologists" that occurred in April, 2012 - a strike-slip fault induced quake of a magnitude of 8.6 on the richter Scale, one of the theories for the great magnitude is explained as


  Another apparent reason for the large magnitude, says Miaki Ishii, a sesimologist at Harvard University, is that multiple faults appear to have slipped almost simultaneously, creating a composite earthquake. “If it was just a single fault activation, it wouldn’t have been large as 8.6. 


An important note the Harvard seismologist makes, that essentially answers your question regarding simultaneous (and near-simulatneous earthquakes), is that observations of this kind of fault activation, which is described as "a cascading failure of faults" are seen in a lot of large earthquakes.