curl: (35) SSL connect error

I saw a similar kind of problem, yet not accessing twitter.
Problem I saw was on RHEL 6.1 -> RHEL 6.4, using curl to access a REST API. curl showed error: curl: (35) SSL connect error

Issue ended up being that our server running the REST API was RHEL 6.6, using a newer version of NSS. Once we upgraded the client NSS version curl was successful.

We upgraded the client from: nss-3.12.10-2.el6_1.x86_64

to: nss-3.16.1-14.el6

According to the source code that error means a time out. Your network connection to that host is most likely faulty. It might also be that Twitter filters out your IP for whatever reason.

It can be a badly configured proxy responding in http instead of https. I've had the case of a proxy always responding in http to the client, no matter the original protocol, when the requested site was blacklisted.

So more generally, this error occurs also when you get a response not TLS encrypted.