I tried upgrading again to Wordpress 4 and the most recent versions of my plugins, because previously I got errors and I didn't have the time to look into it.
Still get the same errors. After upgrading: some pages return a 502 (a minority, but always the same)

502 Bad Gateway
nginx/1.4.6 (Ubuntu)

Previous time I restored a backup of my site (database and website Wordpress 3) and the error disappeared.

Now, I really want to keep the upgrade, but I still get the same error for certain pages. Deactivating W3 Total Cache plugin and all the pages show correctly although not that fast. Looking in the nginx error log file for that site I found this:

2015/06/21 11:12:58 [error] 2549#0: *111218 recv() failed (104: Connection reset by peer) while reading response header from upstream, client: 55.55.55.55, server: blog.johan-mares.be, request: "GET /skeptic/ervin-laszlo-flying-under-the-skeptic-radar/ HTTP/1.1", upstream: "fastcgi://unix:/var/run/php5-fpm.sock:", host: "blog.johan-mares.be", referrer: "http://blog.johan-mares.be/"

For the pages that return a 502 with W3 Total Cache active, the same error is logged each time the page is requested, whether W3 Total Cache is active or not and, I checked previous log files, even before I upgraded.

What could be the cause that certain Wordpress pages log those errors without any effect for the person browsing my site?
Why did it became a problem after upgrading?

Keeping my fingers crossed, but the problem seems solved by upgrading from Thesis 1.8.5 to 1.8.6 theme.
Pages display correctly and no longer "Connection reset by peer" errors in nginx log files.
As to why it only affected certain pages ...