I try to chmod some files and I get the error "disk quota exceeded":

It's true that I don't have that much space remaining in the filesystem where I use chmod, namelynfs.mit.edu:/export/evodesign/beatdb (which is an NFS), either in terms of bytes or in terms of inodes:

But why does chmod need more space?
I use Ubuntu 12.04.
Edit: Some more info following the comments:
1) I don't know the OS of the NFS server (nmap -O nfs-server-ip didn't return anything)
2) The NFS protocol in use by a mounted NFS filesystem is NFSv3:

3) quota -v:

4) I don't have any /var/log/messages logs, I do have some /var/log/syslog logs but nothing looks weird there.
nmap -O nfs-server-ipdidn't return anything) and I'm not sure how to determine which version of the NFS protocol is in use by a mounted NFS filesystem.quota -vto see if you're over quota for that filesystem. You may need to do this when logged in as the owner of the files in question rather than as root. It's possible that your server rejects all writes to files or to file attributes when the file owner is over quota.