Not automatically. The thing about lossy compression -used in most video files- is that once you've compressed the file, the data that was thrown out is gone. There is no way to reconstruct it. The theory behind this technique is that the absence of the missing data won't be noticed by the human eye or ear, but if the video is poorly encoded (or if it was captured from a poor source to begin with) then it becomes all too easy to notice.
The only way to increase the quality of the video at this point, therefore, would be to add in new data. You can use video-editing software to do that, but it's a very long, painstaking process. You may have heard of 'digital remastering' on DVD releases of old films; this is basically what that term means. People get paid to do that as full-time work, and even then it takes a very long time.
If you have some way of re-capturing the video, that would almost certainly be easier than trying to edit the file. Of course, sometimes that's not possible; shooting an event that only happens once can only be done once. Is it possible in your case?