A simple test would be to
telnet $ssh-host $ssh-port. You should see something like the following almost immediately if it worked:
$ telnet sshserver.example.com ssh
Connected to sshserver.example.com.
Escape character is '^]'.
If it takes a while or you get an abrupt failure then it is either not running on the remote server, or you got the port wrong, or it is blocked somewhere between the shell you're running and the remote sshd.
Update based on your answers:
The problem is probably not your university network. Most likely (hopefully) you have something sitting between your home desktop and your ISP (ie your router or WiFi AP etc). That device does not know that you want to allow the inbound SSH connections from the internet. You want to:
1 add a TCP forwarding rule to route an inbound TCP port to a pre-defined destination (your desktop's IP address at port 22). I strongly suggest you pick a big number (larger than 1024 but smaller than 65535) as the port. Your rule will look something like "forward from inbound port 3299 to desk_top_ip port 22"
2 When you are not at home and using your laptop you will have to ssh to your public IP address (not the same as your desktop IP address). You can find your public IP most easily by going to a site like http://whatismyipaddress.com/. It probably doesn't change very often.
3 When you are not at home and using your laptop you will have to ssh to the TCP port you specified above (not the default ssh port of 22). So your ssh command, if you use a command line, would look like
ssh my_user_name@my_public_ip_at_home -p 3288