I have a Concept 300, which is similar. When the car is "immobilised" then yes, you turn the key, and nothing happens, no starter motor sound or anything.
Worth noting that if you get in the car but don't actually start it straight away, then the immobiliser gets put back on after about 30 or 60 seconds. Maybe that's what is happening to you?
Next time this happens, press the "unlock" button on the Clifford Fob while the keys are still in the ignition, and try again.
Might also be worth downloading the manual for your alarm, or just reading it. They have a lot of features (too many!)
Cheers
James