In recent months, when analysts and reporters asked Disney executives about the company’s preparedness, the response was unbridled confidence, in particular from Michael Paull, president of Disney Streaming Services. Mr. Paull, a former Amazon executive, runs the part of Disney that was formerly called BamTech. Disney paid $2.5 billion to acquire BamTech, a New York company that handles streaming for Major League Baseball and helped HBO build its video app.

“We’re confident that we’re really ready for scale — that the BamTech platform has been tested under pretty interesting circumstances, including this past Saturday night, when you had hundreds of thousands of people signing up for a pay-per-view event in a very, very concentrated time,” Robert. A Iger, Disney’s chief executive, told analysts on an earnings-related conference call last week. He was referring was an Ultimate Fighting Championship event.

Disney spent several months testing a trial version of Disney Plus in the Netherlands. The response was “very, very positive — not just the fact that there was an enthusiasm for the service, but we had a good sense about how people were using it,” Mr. Iger said on the call. Analysts estimate that Disney Plus will have eight million subscribers by the end of December; Disney has predicted the service will grow to as many as 90 million members by 2024.

Tuesday’s rollout of Disney Plus in the United States and Canada was heavily marketed across most branches of the company, with buses at Walt Disney World in Florida covered in ads and the host of ABC’s “Dancing With the Stars” urging viewers to sign up for the service during a live broadcast.

Disney Plus is scheduled to arrive in Australia, New Zealand and Puerto Rico on Nov. 19. Countries across Western Europe will gain access in March, Mr. Iger said last week. It was unclear how or if Tuesday’s technical issues will affect those plans.

Some people who had Disney Plus problems on Tuesday seemed understanding of Disney’s learning curve.