Ran into a weird problem trying to lock down our AWS ELBs today, specifically trying to deprecate all support for SHA-1.
Having removed options like AES256-SHA ECDHE-RSA-AES256-SHA (among others) from our external ELB, Chrome and Firefox both refused to connect. Oddly, MS Edge and Safari were still able to connect.
Chrome and FF both reported ERR_SSL_VERSION_OR_CIPHER_MISMATCH.
Based on my best understanding, there's no reason that Chrome or FF shouldn't support larger block sizes for SHA hashing, is there? After all, SHA-1 is deprecated in general. We tried this on several workstations/VMs, each running a different OS (Win10 and OS X El Capitan) and the result was the same for Chrome and FF.
Our certificate was also signed with SHA-256.
Hopefully I'm explaining this reasonably well.
*edit: I found a related list, and it looks like SHA-2 hashing isn't supported by Chrome in conjunction with AES-256.