Is there an in-flight built-in-test in an airplane?
The thing that separates a BIT/BITE (E=Equipment) from continuous fault monitoring is that the continuous fault monitoring simply responds to exceedances of various paramters or values outside of specified limits in real time as they happen (say where an analog voltage signal of a device drops below a minimum of X or exceeds a maximum of Y), whereas a BIT/BITE test is a specific routine where simulated fault signals are injected into a circuit to see if a fault monitor is doing its job. 

BIT or BITE tests of electronic controllers can be categorized into three types:


Start Up BIT: Most common.  A test routine automatically run on power up of the controller.
Initiated BIT: Controllers may have a manually initiated BIT/BITE that can be run any time, but this is usually triggered by a button on the controller itself in the avionics compartment, or through a maintenance access portal in the aircraft maintenance diagnostic system, and usually doesn't have a flight crew initiated facility (but may in some cases).  These are normally intended for troubleshooting.
Continuous BIT: A controller may run an internal self test routine on a continuous cycle while powered up, where a controller has fault detection circuits that can't be allowed to have dormant failures, even for the duration of the flight.