I found that my / was full. This between the other things does not allow to log in from desktop.
So I decided to move a 30GB I had in my home to another disk on the same machine.
I went for:
$ mv BIGDIR /mnt/other_disk_mount_point

than after some time I checked for free space:
df -h

Filesystem Size Used Avail Use% Mounted on
/dev/sda1 578G 556G 0 100% /

files actually moved as I checked. But this free space (te 30GB you can see in the df output) are not considered as available...

before the df output was like
Filesystem Size Used Avail Use% Mounted on
/dev/sda1 578G 578G 0 100% /


What can be done?

May be the files still open so just check whether is there any files open from root partition using "lsof"

I even rebooted and still the same hard drive occupation is shown.