Having both an HOA fee and a condo (C/C) fee is not common, but it is indeed found in various planned communities such as Reston, the Pinecrest in Alexandria, and Penderbrook in Fairfax. A planned community like these typically is a large community with a variety of housing types, such as detached homes, townhouses, and condos (the "flats" type that people usually envision). The community will have shared amenities such as the common green areas, walking paths, swimming pools, clubhouses, tennis courts, etc.; and for all the types of housing, homeowners pay for the use and management of the community amenities. If the owner has a detached home and pays for his own utilities, roof, yard maintenance, exterior painting, etc., his only fee will be that HOA fee. If the owner lives in a condominium, he will have the typical condo fees covering (usually) water/sewer/trash, replacing the roof occasionally, repaving the parking lot occasionally, painting/repairing stairwells and hallways, overall management of just that condo association, insurance for the condo building, etc. So the owner of a condominium in a planned community will pay both the overall HOA fee and the condo fee.
As an abundant example of properties with both types of fees, consider Reston! There is an annual Reston fee of around $500/year. That fee entitles you to use the various Reston pools, recreational facilities, etc.; and it pays for overall management and maintenance of Reston's common areas. If you happen to own a condo in Reston, you will pay a monthly condo fee as well.
When you think about all this, remember that condominium means a form of ownership, not a physical type of property. Condominiums come in all physical types of housing, including "flats," townhouses, and even single-family detached homes (yes indeed--for example, there is a small community of detached homes in north Arlington that has a form of condominium ownership--one home is currently on the market). Also remember that the condo fee is not evil or something to be avoided; it covers services, repairs, utilities, insurance, etc., that the owner of a detached home pays out of his own pocket. A fraction of the condo fee does go toward management services, which the individual homeowner doesn't pay for--except that the individual homeowner is using his own time and energy to manage services himself and arranging for his own roof repairs, lawn maintenance, exterior painting, etc.