Stephanie Russell-Kraft started work in September as a reporter covering the business of law for New York-based Thomson Reuters Corp., a job she says she considered a step up after spending two years at legal newswire Law360.

Within weeks, however, Thomson Reuters asked Ms. Russell-Kraft to leave. Her previous employer, Law360, also in New York, had sent a letter to the company citing a noncompete agreement she signed when she started at the newswire as a 25-year-old, her first full-time job in journalism. The letter, a copy...