Why does my bathroom cold water faucet cut off while in use?
Typically with a blocked pipe, a small flow of water can get past the blockage and equalise the pressure both sides of the blockage. When a far away faucet is opened the water on it's side of the blockage flows out, giving an impression of a good flow for a short while (maybe a few seconds), however after that the flow rate is determined by leakage past the blockage. This can seem like a small trickle or may not be enough to really notice. 

I have had this happen a few times due to sediment gradually accumulating in a filter in a pressure reduction valve. Cleaning the filter solves the problem for a year or so.

I would check all the valves and other devices upstream of the faucet one by one. Blockages often occur at restrictions like these.