To interrupt= to stop someone from speaking by saying or doing something.
To bother = To annoy, worry or cause problems to someone
Exemple: Don't bother your father when he is working.
Disturb= to cause someone to stop what the person is doing.
Exemple: Please don't disturb Jimmy - he's trying to do his homework.
"Interrupting" and "disturbing" someone isn't necessarily annoying which isn't the case with "bothering". In this phrase only "interrupt" and "disturb" are interchangeable, however not "bother".