We investigate clogging of microchannels at the single-pore level using microfluidic devices as model porous media. The process of clogging is studied at low volume fractions and high flow rates, a technologically important regime. We show that clogging is independent of particle flow rate and volume fraction, indicating that collective effects do not play an important role. Instead, the average number of particles that can pass through a pore before it clogs scales with the ratio of pore to particle size. We present a simple model that accounts for the data.