There are several problems with Finlock guttering.
The joints can leak,as already mentioned. The repair systems used to line (or reline)vary a lot depending on the materials used..like everything,you get what you pay for.Lead, properly applied,would be my recommendation,then running down to bitumen or heaven forbid,mastic gun man..Al at lowering cost/effectiveness/longevity.
Often properties with Finlock gutters date back to when galvanised metal window frames were used.These provided partial support for the heavy gutters.Many have been replaced by pvc u frames and few of these allow for the extra weight and settle ,causing cracking at the joints,leading to leakage.
The concrete,running laterally from cold outside to warm inside can lead to condensation and mould growth.
I no longer have my surveying textbooks and can't remember if the gutter sections have steel reinforcement which could corrode......
Apart from that, there's not much wrong with them.