With conventional loans, once the LTV (loan to value) ratio = 80%, the lender CAN drop the PMI fee. Once the LTV = 78% or less, the lender MUST remove the PMI from your loan payment.
The FHA insures loans itself. There is a fee for this service. This fee can be and is removed from your loan payment at a specific time, based on LTV and age of loan and payment history.
FEDERALLY insured loans, by definition, do not require PRIVATE mortgage insurance, although semantically I suppose you could be corrrect in thinking that "an insurance fee is an insurance fee is an insurance fee."
But anyway you look at them, they DO get removed. Lenders need to feel secure before they'll make a loan. They have statistically/actuarially found that they have sufficient security if the LTV ratio is 80% or less. Only when the LTV ratio is greater than 80% do they require insurance.
However, many lenders will not voluntarily remove the extra fee from your loan payment even though they have cancelled the insurance with the PMI firm. It is often up to you to inform them of the status of your loan. Talk with your lender about this. If you have lived in your home for several years, and you are paying PMI, this is definitely worth looking into.