The diagram below shows in more detail the best times to repair a tyre and the best times to replace a tyre.
Most punctures are made by nail holes or cuts of some kind. If it is up to 6mm then it can be repaired by trained technicians. However, the damage needs to be confined to the tread.
When looking at the diagram the green area is the fixable zone. The red area is the replace zone.
A puncture repair job is a lot cheaper and far less hassle. If you suspect that you have a puncture it is best to bring it in right away.
The longer you leave it the worse it can get and instead of a repair job, you might have to replace a tyre.
There can be a lot of danger if left unchecked as well, your tyre could blow out while driving. Which could cause more damage to the car but possibly cause an accident.