This issue seems to have reappeared. We're using v.1.626 and have also reproduced this issue with the latest LTS 1.609.2 as well.
1. Login with valid credentials,
2. 404 "Not found" page displays,
3. Remove session ID URL parameter and hit enter,
4. Jenkins home displays for the credentialed user.
This behavior is the same whether we invoke Jenkins through our Nginx reverse proxy (http://192.168.41.1/jenkins) or bypass the proxy and go directly to Jenkins running on http://192.168.41.1:8080/jenkins.
AWS Ubuntu 14.04 EC2 instance
VPN access only (192.168.41.1 Nat'd to the instance's private IP)
Nginx 1.4.6 (on port 80 with reverse proxy to Jenkins running on WildFly 8080)
Oracle Java build 1.8.0_60-b27