curl: (35) SSL connect error

I am trying to make a cURL HTTPS request to the Twitter API and I am getting this error:

[root@webscoming httpdocs]# curl -v https://api.twitter.com
About to connect() to api.twitter.com port 443 (#0)
Trying 199.16.156.199... connected
Connected to api.twitter.com (199.16.156.199) port 443 (#0)
Initializing NSS with certpath: sql:/etc/pki/nssdb
CAfile: /etc/pki/tls/certs/ca-bundle.crt
CApath: none
NSS error -5990
Closing connection #0
SSL connect error
curl: (35) SSL connect error

I have tried this from another server and it works as expected. What does that error mean and how do I resolve it?

My server is running CentOS version 6.5.

Please note: I can correctly connect to https://encrypted.google.com but unsuccessful with https://api.twitter.com.

I saw a similar kind of problem, yet not accessing twitter.
Problem I saw was on RHEL 6.1 to RHEL 6.4. curl showed the 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