Anecdotally?
Only Bing can directly answer your question -- but I believe I can indirectly answer it. I think your experience also answers it, although indirectly.
Another way to ask that question is does Bing and other search engines penalize sites for usability and readability. Isn't the common knowledge for that question yes they use usability and readability as a factor? But does that include readability for those with sight problems?
The semantic usage of the H tags are a table of contents for screen readers. And, although there was consideration to add more than one h1 tag to a page, (more specifically two), no software supports that usage ... so the standard is one h1 tag.
As for the usage of the table of contents, (h tags), for search engines; Search engines do look at it; as well as looking at say BOLD text, but if all text is bold then bold means nothing ... likewise if all h tags are h1, then h1 means nothing. Anecdotally, I've experienced the same when all information is equally made important.
The best practice is within CSS have the display rule be
h1, .h1 { /* use .h1 class to display subtitles in h1 style */ }
and use one h1. Just like web pages use one title.
On a person note; I use TOC to determine what a book is about on a regular bases ... although i don't have usability software to do the same on the internet and most sites don't use TOC properly ... it would be nice if web sites considered using the h tags semantically. And, if normal browser supported looking at the TOC directly as a sidebar or something for naviagation.