Can A Dirty Air Filter Make You Sick?
of course it can
More than you know.
i would like to point out however though as a air filter becomes dirty it actually becomes harder for anything to pass through the filter this includes more dirt the reason we replace them is eventually they become so clogged that the air flow is reduced to the point that it needs replacing. Also unless you have serious allergies to anything in the air then as long as nothing mold like is growing on the air filter then its not going to get you sick. however if its not installed properly or it is pumping a lot of dust and dirt into the air in your house it can cause irritation to the respiratory track which can lead to infections because of micro tears from inhaling an excessive amount of coarse materials that the bodies natural defenses couldn't keep up with but you would probably be able to see that amount of dust and dirt in the air on any given day so its doubtful that would go unnoticed
It probably depends on what the dirt is. Like "plain" dirt will not, but say mold or anything like that sure could.