CBS has not yet confirmed nor denied that "Person of Interest" is getting a fifth season but the powers that be are pointing to it eventually happening. 

Executive producer Jonathan Nolan already announced that the season 4 finale would end on a cliffhanger that would lead directly into season 5, as cited by Seriable. Also, when asked at the PaleyFest panel regarding the chances of a fifth season, Nolan stated that "we're feeling very good." 

Seriable notes that last April 14, Nolan continued to talk about the show as if it was common knowledge that there is going to be a fifth and sixth season down the line. The executive producer continued to use phrases such as "next season" when discussing plot lines and the fourth season's cliffhanger finale. 

Ratings for the series, as posted by TV Series Finale, are within the 8- to 10-million-viewer range for the 18-49 age demographic, garnering an average rating of 1.55. Most shows that get renewed require a 4- to 6-million-viewer range for this age demographic and "Person of Interest" is well above that line. As cited by Cartermatt, CBS is experiencing some rating issues with its other shows. Excluding "NCIS: Los Angeles," the network doesn't generally earn anything better than an average 1.30 rating for its shows. 

Moreover, according to a report from SpoilerTV, a fan visiting the set was informed by one of the crew members that the show was officially being renewed for a season 5. The fan posted the information via social media but CBS has not made any comments regarding the leaked information. 

Conversely, Cartermatt explained that one of the main reasons the show may get cancelled is because it is a Warner Bros. show on the CBS network. It reasoned that networks tend to favor in-house projects and this makes "Person of Interest" a lower priority compared to the other series that the network has up for renewal.