[HTC Sensation] rhd-android:
I have some additional data / clues to add to the mix here:

As I mentioned previously, no problem occurs over WIFI, but the problem occurs if I switch to mobile data. So I took a logcat of the failure (error 107 when trying to retrieve https://www.twitter.com) over mobile data, followed by another logcat of the problem not occurring over wifi.

The problem occurring (mobile data):

W/chromium( 1701): [WARNING:http_stream_factory_impl_job.cc(1035)] Falling back to SSLv3 because host is TLS intolerant: twitter.com:443
W/chromium( 1701): [WARNING:http_stream_factory_impl_job.cc(1035)] Falling back to SSLv3 because host is TLS intolerant: twitter.com:443
W/chromium( 1701): [WARNING:http_stream_factory_impl_job.cc(1035)] Falling back to SSLv3 because host is TLS intolerant: twitter.com:443
W/IInputConnectionWrapper( 1701): getExtractedText on inactive InputConnection
W/ThrottleService( 348): unable to find stats for iface rmnet0
E/chromium( 1701): [ERROR:ssl_client_socket_openssl.cc(803)] handshake failed; returned -1, SSL error code 1, net_error -107
E/chromium( 1701): [ERROR:ssl_client_socket_openssl.cc(803)] handshake failed; returned -1, SSL error code 1, net_error -107

Everything working fine (wifi):

(I redacted a few portions that seemed to be unique IDs in URLs. I replace the IDs with "--redacted--")

W/chromium( 1701): [WARNING:http_stream_factory_impl_job.cc(1035)] Falling back to SSLv3 because host is TLS intolerant: twitter.com:443
W/chromium( 1701): [WARNING:http_stream_factory_impl_job.cc(1035)] Falling back to SSLv3 because host is TLS intolerant: twitter.com:443
W/chromium( 1701): [WARNING:http_stream_factory_impl_job.cc(1035)] Falling back to SSLv3 because host is TLS intolerant: twitter.com:443
W/chromium( 1701): [WARNING:http_stream_factory_impl_job.cc(1035)] Falling back to SSLv3 because host is TLS intolerant: twitter.com:443
W/chromium( 1701): [WARNING:http_stream_factory_impl_job.cc(1035)] Falling back to SSLv3 because host is TLS intolerant: twitter.com:443
W/chromium( 1701): [WARNING:http_stream_factory_impl_job.cc(1035)] Falling back to SSLv3 because host is TLS intolerant: twitter.com:443
I/dalvikvm( 1701): Jit: resizing JitTable from 4096 to 8192
D/dalvikvm( 1701): GC_CONCURRENT freed 6169K, 68% free 5376K/16643K, paused 8ms+12ms, total 185ms
I/chromium( 1701): [INFO:CONSOLE(17)] "Revision: --redacted--", source: https://mobile.twitter.com/ (17)
D/ChromeBrowserSyncAdapter( 1701): no delayed sync
I/chromium( 1701): [INFO:CONSOLE(0)] "Creating Application Cache with manifest https://mobile.twitter.com/cache/manifest", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Checking event", source: (0)
W/chromium( 1701): [WARNING:spdy_session.cc(1117)] Received data frame for invalid stream 13
I/chromium( 1701): [INFO:CONSOLE(463)] "Loaded: 1787", source: https://mobile.twitter.com/ (463)
I/chromium( 1701): [INFO:CONSOLE(462)] "Started: 1855", source: https://mobile.twitter.com/ (462)
I/chromium( 1701): [INFO:CONSOLE(53)] "appcache: checking", source: https://mobile.twitter.com/ (53)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Downloading event", source: (0)
I/chromium( 1701): [INFO:CONSOLE(53)] "appcache: downloading", source: https://mobile.twitter.com/ (53)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (0 of 12) https://ma.twimg.com/twitter-mobile/--redacted--/html5/framework/core/assets/m_spinner_black.png", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (1 of 12) https://ma.twimg.com/twitter-mobile/--redacted--/html5/applications/m5/views/core/timelines/assets/dotted-pattern.png", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (2 of 12) https://mobile.twitter.com/i/templates/m5", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (3 of 12) https://mobile.twitter.com/", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (4 of 12) https://ma.twimg.com/twitter-mobile/--redacted--/html5/framework/core/assets/twitter_logo.png", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (5 of 12) https://ma.twimg.com/twitter-mobile/--redacted--/html5/plugins/defer/pullrefresh/assets/pulltorefresh_bg.png", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (6 of 12) https://ma.twimg.com/twitter-mobile/--redacted--/html5/applications/m5/assets/sprite_mobile.png", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (7 of 12) https://mobile.twitter.com/assets/m5_defer.js", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (8 of 12) https://ma.twimg.com/twitter-mobile/--redacted--/html5/framework/core/assets/m_spinner_white.png", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (9 of 12) https://ma.twimg.com/twitter-mobile/--redacted--/html5/applications/m5/views/core/profile/assets/bg_profile_empty.png", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (10 of 12) https://mobile.twitter.com/assets/m5_defer_ssl.css", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (11 of 12) https://ma.twimg.com/twitter-mobile/--redacted--/html5/applications/m5/views/core/timelines/assets/list-gap.png", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Progress event (12 of 12) ", source: (0)
I/chromium( 1701): [INFO:CONSOLE(0)] "Application Cache Cached event", source: (0)
I/chromium( 1701): [INFO:CONSOLE(467)] "appcache: cached", source: https://mobile.twitter.com/ (467)
W/chromium( 1701): [WARNING:http_stream_factory_impl_job.cc(1035)] Falling back to SSLv3 because host is TLS intolerant: twitter.com:443