Only a non-expert opinion here but it seems to me you would be ok doing it the other way round but not this way.
For example, if you had a feed in 4mm, running something else from that at 1.5mm is fine assuming the load is pulling an amperage which the 1.5mm is rated for and that you used a fused spur. A fused spur needs to be used since if there's a fault the 1.5mm needs to spike high enough that the 4mm's breaker will detect the fault and trip the circuit off. (So something wired in 1.5mm would usually use a breaker rated between 6 and 16a, but 4mm would use one at 32a so a fault on the 1.5mm needs to be able to trip the 32a breaker the 4mm is being fed from otherwise the cable could heat up and cause a fire.)
The other way round, so connecting 4mm into a 1.5mm feed, the appliance that needs to use the 4mm cable is going to pull much more power, which has to come through the 1.5mm and it's probably not rated for it.
It's all about the rating of the cable compared to what the load is trying to pull through it, so try to think in terms of rating instead of physical size - a 16mm cable is just a conductor that can carry a certain amount of power, a 1.5mm is one that can carry a lot less.
If i'm wrong someone will correct me but this is how i understand it.