Teel Lidow couldn’t quite believe the numbers. Over the past few years, the nation’s largest telecom companies, like Comcast and AT&T, have had a combined 330 million customers. Yet annually an average of just 30 people took the companies to arbitration, the forum where millions of Americans are forced to hash out legal disputes with corporations.

Mr. Lidow, a Silicon Valley entrepreneur with a law degree, figured there had to be more people upset with their cable companies. He was right. Within a few months, Mr. Lidow found more than 1,000 people interested in filing arbitration claims against the industry.

About the same time last year, Travis Lenkner and his law partners at the firm Keller Lenkner had a similar realization. Arbitration clauses bar employees at many companies from joining together to mount class-action lawsuits. But what would happen, the lawyers wondered, if those workers started filing tens of thousands of arbitration claims all at once? Many companies, it turns out, can’t handle the caseload.

Hit with about 2,250 claims in one day last summer, for example, the delivery company DoorDash was “scared to death” by the onslaught, according to internal documents unsealed in February in federal court in California.