In theroy every bulb should have a built in fuse so if when it fails there is ionisation inside the bulb (The pop and flash as it blows) the fuse in the bulb should rupture rather than taking out main fuse. Fuses can be designed to work quicker than a trip and semi-conductors are protected with a fuse not a trip.
However this is theroy in practice the fuse inside the bulb is often missing. When means when on blows it often takes out the trip or fuse in the consumer unit.
The consumer unit should have a 5A fuse or 6A trip as a maximum as the ceiling rose is only rated 5 or 6 amp and is used as a junction box so in most cases this limits the maximum size. If ceiling roses were not used the regulations allow up to 16A and as a result I have seen many mistakes with larger than permitted fuse or MCB.
The result from all this is often dimmer switches fail when bulbs blow because either they are just not protected or the protection is just not good enough to save the semi-conductors inside the unit.
So most likely is the dimmer has blown but you should also look before replacing to ensure the correct protection is in place in the consumer unit. In the main a B6 MCB is used for lighting. It has been a problem with the faster acting MCB that they will open when one bulb blows and often a C6 or D6 is fitted to try and stop this. May be OK with standard switches but not when using a dimmer.
The B, C, and D mean the magnetic part of MCB works at 5, 10 or 20 times the thermal rating of the device so a B6 will allow 30A to pass clearly even this is a little high for a humble dimmer switch. With 60A (C) or 120A (D) the dimmer will clearly fail.
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