700c rims have a 622mm bead diameter. The "700c" is not an actual engineering measurement. It's merely a class of wheels who share a 622mm bead diameter and who's tires generally fall within the gross diameter of 700mm.

Borrowed from wikipedia:

"Sizes[edit]

Bicycle rims and tires came in many different types and sizes before efforts were made to standardize and improve wheel/tire compatibility. The International Organization for Standardization (ISO) and the European Tyre and Rim Technical Organisation (ETRTO) define a modern, unambiguous system of sizing designations and measurement procedures for different types of tires and rims in international standard ISO 5775. For example:

• For wired-edge tires the ISO designation lists the width of the inflated tire and the diameter with which the tire sits on the rim, both in millimeters and separated by a hyphen: 37-622

• For rims the ISO designation lists the rim diameter (where the tire sits) and the rim's inner width, both in millimeters and separated by a cross, along with a letter code for the rim type (e.g., "C" = Crochet-type): 622x19C

In practice, most tires (and inner tubes) sold today carry apart from the modern ISO 5775-1 designation also some historic size markings, for which no officially maintained definition currently exists, but which are still widely used:

• an old French tire designation that was based on the approximate outer diameter of the inflated tire in millimeters: 700×35 C.
• an old British inch-based designation: 597 mm (26 × 1¼), 590 mm (26 × 1⅜, which is the most common), 630 mm (27 × 1¼), and 635 mm (28 × 1½)

Which designation is most popular varies with region and type of bicycle. For a comprehensive equivalence table between old and new markings, see the ISO 5775 article, the table in Annex A of the ISO 5772 standard, as well as Tire Sizing by Sheldon Brown.
Most road and racing bicycles today use 622 mm diameter (700C) rims, though 650C rims are popular with smaller riders and triathletes. The 650C size has the ISO diameter size of 571 mm. Size 650B is 584 mm and 650A is 590 mm. 650B is being promoted as a 'best of both worlds' size for mountain biking.[SUP][36][/SUP] Most adult mountain bikes use “26 inch” wheels. Smaller youth mountain bikes use 24 inch wheels. The larger 700C (29 inch) wheels have enjoyed some recent popularity amongst off-road bicycle manufacturers. The formerly popular (27 inch) wheel size is now rare. These rims are slightly larger in diameter than 700C wheels and are non-compatible with bicycle frames and tires designed for the 700C standard.
Children's bicycles are commonly sized primarily based on wheel diameter rather than seat tube length (along the rider's inseam) dimension. Thus, a wide range of small bike wheels are still found, ranging from 239 mm (12 inches) diameter to 400 mm (18 inches).
Wheel rims also come in a variety of widths to provide optimum performance for different uses. High performance road racing rims are narrow, 18 mm or so. Wider touring or durable off-road tires require rims of 24 mm wide or more.[SUP][37]"

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To the best of my knowledge there is no standard min/max with regard to tires for a specific internal rim width. If you were to look out in my garage you would find tires as wide as 57mm still mounted to a 19mm rim. But, at only 3.5bar vs my 25mm tire which are also mounted to a 19mm wide rim but inflated to 8 bar.