Is it possible to get a better quality by increase the frame rate, even if I don't have the raw footage?
If you are really talking about frame rate, then if you tried to invent new frames while maintaining the speed of playback, your quality level would go down (likely significantly).  

If you didn't mind it looking like someone was watching a fast forwarded version of your video, increasing the frame rate should have some increase in quality as long as the frame rate isn't too high already due to the dithering impact over time resulting in better apparent quality (this is why still frames of video don't look as good as playing the video.)

If you actually meant bit rate (the amount of data being used to encode each frame), then no, all that would happen on the re-encode is that your existing problems would be preserved with less additional quality loss, but it is impossible to restore quality by increasing bitrate.  The information is already lost.  In fact, if you do highly lossy compression more than once, the quality loss starts growing exponentially as the random noise destroys the effectiveness of compression algorithms.

This is why renders are run from the highest available source so that the compression can make the best use of real information without having to waste bits on trying to compress noise.