I discovered how low most of my coworkers are. In total lack of integrity they cannot admit that they falsely accused one of the guys, not to mention about defending him, right Mr. Integrity Man?
Don't worry I feel the shame for you. Enjoy standing tall and being right, because you are standing alone and more alone you will be unless you realize this.

The poor guy is a programmer, yet he is blamed for a company mail server running out of hard drive which is entirely network's administrator's fault.

Next thing, I guess, will be to blame my Web Designer for future, misconfigured firewall for our WebSite server at the hosting facility. After all, a firewall configuration "is an essential component for the it to run and if we are using you should know how it works!" (as my boss is saying) In another words my Web Designer is surely incompetent.

Any advise for the staff?
Sure: Know it all, from accounting, thru customer service, be a superior programmer in at least 5 languages, configure Web servers, email exchange servers, network administration and security, fax, ftp, all the possible services and their effect and everything else, know all the Microsoft bugs and fixes and then and only then, maybe my boss will consider keeping you.