We have studied the rate and gain limits of diamond-coated Microstrip Gas Counters (MSGCs) and Micro-Gap Counters (MGCs) when combined with various preamplification structures: Gas Electron Multiplier (GEM), Parallel-Plate Avalanche Chamber (PPAC) or a MICROMEGAS-type structure. Measurements were done both with X-rays and alpha particles with various detector geometries and in different gas mixtures at pressures from 0.05 to 10 atm. The results obtained varied significantly with detector design, gas mixture and pressure, but some general features can be identified. We found that in all cases, bare MSGCs, MGCs, PPACs and MICROMEGAS, the maximum achievable gain drops with rate. The addition of preamplification structures significantly increases the gain of MSGCs and MGCs, but this gain is still rate dependent.