One of my LinkedIn contacts was recently admitted to several graduate schools as well as a prestigious international fellowship program, and over the course of a few weeks, she posted the acceptance letters as she received them before finally announcing the PhD program she'd chosen to attend. I've seen similar announcements from other contacts, and I've often wondered whether or not such posts are helpful to one's professional standing.
In my opinion, a letter of admission from grad school or a fellowship offer is a reflection of your accomplishments but not an accomplishment in its own right, and thus doesn't belong on your resume.
However, on LinkedIn, space is unlimited and posts recede quickly into the bottom of the feed, so I wonder if it might be more acceptable to post about declined opportunities there. For those just starting out in their career, there's pressure to provide "hard proof" of their competency. If you got both a job offer and a grad school acceptance, you might want to indicate that you were prepared for the one even though you chose the other, particularly if they are in different fields. And I have seen cases where the poster was admitted to a very prestigious graduate program but (I can infer) chose to attend another for financial reasons.
Thus, my questions are
How do recruiters perceive it when they visit someone's LinkedIn feed and see posts about declined opportunities?
Is there an acceptable way to share that you have been offered a competitive professional opportunity that you will not, or cannot, accept?