Did your partition fill up because you installed too many packages or because of downloads or expanding log files? It's possible that uninstalling some packages now just delays the same problem occurring again later on.
It might help to check that your log files being rotated and re-used so that there's a limit on how much space they take up. Could there be an auto updater that doesn't delete the packages it downloads?
You can check where the high usage us by running the following:
If it showed that very high usage was in /usr, you'd then run:
And so on until you found where the files are.
How much space you require to boot depends on what you are running. 600MB (5.8G - 5.2G) sounds like a lot, but for all I know you have a file for swap space instead of a partition and are running something like Oracle on a box with low RAM...