After I got my Ph.D. in statistics in the 90s, if I hadn't been rejected at the three universities where I made the short list and was invited for interview and job talk (offers went to different persons on the short lists…), I would not have been free to accept the offer of a postdoc here at HSPH Dept of Biostatistics from Louise Ryan! I am still in Dept of Biostatistics, now Sr. Research Scientist, and it has been (and is) a great experience and excellent place to be a biostatistician. So without those rejections, I wouldn't be here!
P.S. I may still have my favorite rejection letter, from Polaroid after undergrad – it was a beautiful color picture card. While essentially still a 'form letter' it was very well done.