I think this is child and event dependent. DS did math pentathlon in kinder and ended the event in tears because he only won 3 out of 5 matches. This is not a high stress competition, but I think he was a little young. The next competition was in Destination Imagination (DI) last year (second grade) and it was a great experience for him. They didn't win, and I never even saw the score sheet. The kids totally didn't care. It was a great team competition.
This year (third grade) he tried out for the UIL (University Interscholastic League - it's a Texas thing) Spelling competition at his school, made the team and went on to win a medal at the district competition. It was a great experience for him to have to work hard over a long period of time to be successful and to compete with other students who are the best at their schools. At his age I did have to plan the studying for him so that he worked a little each day (15-20 minutes) so that it didn't become overwhelming for him. I feel like that kind of event would have been a bit too much pressure a few years ago, but was great for his current stage of development.
He is doing DI again this year, and I doubt they will place as some teams are super committed and only admit gifted students (our school accepts all students), but it has been fabulous learning experience for him again this year.