The tires should have detailed specs that indicate the rim size required, in both diameter and width - as long as you're staying within the correct rim size, you will probably be fine, keeping in mind what everyone else said. Mounting a tire that requires a narrower rim can be deadly.
For example, the Michelin Defender P175/70R13 requires 4.5"-6" width rims, and is recommended for the Honda Civic DX, while the Firestone Winterforce P155/80R13 requires 4"-5" width rims.
Not that I'd recommend a Winterforce tire to save on fuel (Firestone rates it as a "5" for fuel efficiency), I was just using it as an example to show the difference in rim width requirements.
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