It would be nice to have more details:
column dimensions, particle size,type of packing, flow rate, pressure, and the amount of time it takes to overpressure.
In the absence of data, my guess is that air, per se, is unlikely to cause overpressure. Liquids are somewhat compressible, and an HPLC system doesn't pressurize instantaneously, so a partial blockage can result in a gradual pressure rise. Check with the manufacturer to find out if you can back-flush the column. If so, try that (may get rid of junk on the inlet frit).
Some columns with polymer-based packings have maximum flow rate limits. If the flow rate is exceeded, the particles presumably start to deform and the pressure rises in the manner you describe.