Which resource is failing? And what errors do you get in the engine log/agent log/system log? Chances are that a monitor is trying to write a temporary file to the root file system, and that is causing the behaviour, but without more information, it's impossible to tell.
A fault occurs when a resource goes offline unexpectedly. In other words, the monitor reports that the resource was online, then it returns an offline state, and VCS wasn't involved in that change. An example would be hares -offline involves VCS changing the state.
If that resource is marked as Critical, the default behavious is to failover the service group. Under normal circumstances, the offline is reported because something has happened to the resource, but if the monitor itself fails, that can have the same effect.
In that case, the log files will give the cause. Any fault or failure should be investigated
The root disk filling, won't, by itself, necessarily cause a failover.