if you had a loose or burnt connection, or two, then no. there would be no overcurrent or short circuit to trip the MCB. But connection would be lost. If some of the sockets on the circuit are working and some are not, then it is quite likely a bad connection inside a socket. If they are on a ring then you have at least two faults. It might also be cable damage, typically from a nail or screw in a wall piercing a cable (this can be very dangerous)
Do you feel confident to trace the fault yourself? Have you got a multimeter? (if not, go up to the high street and buy one. it should cost £10 or less)
You will also need a medium and a small electrician's screwdriver, and probably a couple of metres of green and yellow sheathing which you will find in the electrical tools section.
If you are not confident to trace the fault yourself, ask around friends and neighbours for a recommended local electrician. Before asking one to come round, ask if he is a member of a self-certification scheme; which one; how long he has been a member, and what trading name he is registered under (this enables you to check his credentials). this is not a foolproof way of finding a good electrician, but it is a good start. If he is not qualified to self-certify then he is probably not a good and up-to-date domestic electrician.