If you do not find anything more automated, here is a workaround solution that will work at least for [`.srt` files](https://en.wikipedia.org/wiki/SubRip#SubRip_text_file_format):

You can use [Nikolaj Lynge Olsson's *Subtitle Edit*](http://www.nikse.dk/subtitleedit/) to *first* adjust each file individually to the video - you do *not* have to do this line by line; for each file, just pick one of the first lines and one of the last lines and manually adjust their time of appearance and duration. *Subtitle Edit* can then adjust the rest of the file for you.

Once that is done, you can use the *Tools* -> *Append subtitle* command to join all files in *Subtitle Edit* and save them as a complete subtitle file.

*I am not affiliated with that project. However, I have often used the program for synching subtitles with little effort.*