Some things you need to verify:
In the Admin Panel, is "Allow Attachments" and "Accept emailed files" checked?
Is the attachment directory path correct (it must be an absolute path, not relative)?
Is the directory writable?
Are the files that are being attached allowed in the "Accepted File Types" field?
Are the files not too big?
Did you try uploading an attachment when opening a ticket via the website (so not by email) and did it work (this way you can rule out some things)?
If everything is alright and it is still not working, try enabling error_reporting and look at the log files to see what's going wrong. The last option is to follow the steps in the code and try to debug it from there.
Good luck.