Plex tends to do a good job of loading local subtitle files (usually .srt) and it can even automatically load subtitles for you [2]. But sometimes, it will not load a file that is in your directory with your video file.

The solution is likely a quick fix: add the Language code to the subtitle file. From Plex’s support [3]

Movie_Name.avi
Movie_Name.[Language_Code].srt

Example:

/Movies 
     Avatar (2009).mkv 
     Avatar (2009).en.srt

References

1. http://support.plex.tv/articles/categories/using-subtitles/
2. https://support.plex.tv/articles/200288597-fetching-internet-sourced-using-your-own-subtitle-files/
3. https://support.plex.tv/articles/200471133-adding-local-subtitles-to-your-media/