At what point does a stream become a river?

In Wisconsin, there is no legal distinction between a river or a stream.

http://dnr.wi.gov/topic/fishing/questions/lakeriv.html

In general, brooks, streams, and creeks are similar terms.

A river is a stream of water of considerable volume, which travels downhill (from higher altitudes to lower altitudes due to gravity).

Rainfall will seep into the ground or become runoff, which flows downhill into rivers.

Typically, creeks and brooks are smaller than streams while rivers are much larger than streams and other flowing water sources.

Little creeks or streams merge to form small rivers then become medium-sized rivers.

These rivers may be tributaries of a large river which can eventually flow into the ocean. This interconnection between streams and rivers form watersheds.

http://www.differencebetween.com/difference-between-stream-and-vs-river/