When Does "The Big Sick" (2017) Take Place?
"The Big Sick" is a romantic comedy film written by Emily V. Gordon with Kumail Nanjiani that released June 23rd, 2017. So when is "The Big Sick" set?
It takes place in the year:
We know this because Kumail Nanjiani uses Emily Gardner's thumbprint to access her iPhone smartphone and this feature was added on September 10th, 2013 AD. So it is very likely after September of 2013 AD.

2016 AD
We know this because Kumail Nanjiani uses Emily Gardner's thumbprint to access her iPhone smartphone and this feature was added on September 10th, 2013 AD. So it is very likely after September of 2013 AD.
Furthermore, at the end in the new version of Kumail's one-man show she says, 'The only time I've cried that much is during the first 15 minutes of "Up".' referring to the animated movie "Up" (2009), which released on May 29th, 2009 AD in the real world. So it must be after May of 2009 AD.
More specifically, when Kumail listens to old voice mail recordings from Emily, one is labeled "June 10, 2016". In the same scene his iPhone smartphone lock screen clearly reads "Sunday, June 12" (though it says the same later during a scene several days afterward). Also, near the end a poster for Kumail's one-man show reads "Playground Theater, June 20th" (though two shots later we see a Facebook event page for the show that reads "Tuesday, March 10"). So it is probably June of 2016 AD for roughly the last half of the narrative.
[HOWEVER, it has also been reported that the plot is based on Gordon and Nanjiani's own romantic/medical experiences. The real-world events of their courtship, medical disaster, reunion, and move to New York City occurred in 2006 AD - 2007 AD. Their first meeting was in 2006 AD, her illness began in March of 2007 AD, the intense period shown on screen lasted just a little more than 8 days and they moved together to New York within 3 months of the middle of the story, so in May or June of 2007 AD. This HistoryVsHollywood.com entry examines its exact setting.]
Additionally, there is a short epilogue that is unplaceable.