Are you using a download accelerator?
Do you know who's running the server you're trying to connect to?
If you use Firefox you can try FireFTP to see if it is Filezilla at fault.
The number of allowed connections is a server side setting - so trying another FTP program probably won't cure your problem as it's just going to ask for another connection from your IP address and be denied.when trying to transfer files it keeps saying
Status: Delaying connection due to previously failed connection attempt...
Response: 421 Too many connections (8) from this IP
Error: Could not connect to server
I like webdrive - sets up your ftp connection to act like a network drive and you can then work on files as if they are on your pc rather than download edit save upload.