They generally don't blow at the rated current. I've reliably used a 13A fuse on something that draws 16A for a while when starting up and never had a problem. Obviously RCDs etc are more accurate, but wired / cartridge fuses are always under-rated, in my experience.
Most fuses are slow blow - they will allow somewhat over the rates power draw for a while before blowing. Many things (motors) take a lot more than rated current for a few seconds and this is perfectly safe so the fuse shouldn't blow. The purpose of the fuse is to prevent a house fire, if your wires are good for 13 amps that means at 100% duty cycle, you can draw 16 amps for "a while" before the wires get hot enough to burn the house down, so long as you drop under 13 amps for "a while" so the wires cool down and never get hot enough to start a fire the fuse is working.
Note that the above is about safety. If you are going to draw 16 amps you should stick to a circuit rates for that, as it will have less resistance and thus waste less energy.
