Do brake pads wear (i.e. thickness) over time affect the stopping distance
Brake pads and brake shoes are engineered with sufficient thickness to last a specific number of miles combined with type of use. The brakes will provide consistent braking ability over the majority of the life of the pads/shoes. Some manufacturers suggest to brake gently when operating brakes which are new (from the showroom) or which have had the pads recently replaced.
It is common for brakes to provide better stopping power after this conditioning period. I wanted to say breaking-in period, but it felt wrong. Car and Driver magazine frequently reports on long term testing of reduced stopping distance when compared to the first test performed on their long-term vehicles.
Today's designs incorporate a warning device, usually a metal tab or flange that will contact the brake disk before the thickness is reduced to a dangerous level.
If the brakes are not abused during the early period and not abused during use (hard braking, repeated hard braking causing overheating) it is likely you won't notice a difference until they are replaced or the warning squeal appears.