Internet Explorer 6 supports TLS 1.0. However, for some reason that escapes me, https://tls-v1-0.badssl.com:1010/ (which is supposed to test whether a browser supports TLS v1.0) fails to open in Internet Explorer 6. And it fails to open not with "Cannot establish a secure connection.", but with "Cannot find server or DNS error.". Firefox 52 in that same virtual machine can open that web-page. What is going on?
I believe it's because that version of Internet Explorer does not support Server Name Indication, which is part of the TLS protocol which allows a secure handshake to be established when multiple domains with different certificates are hosted at the same IP. If badssl.com is hosted on shared hosting, then I think this explains your issue.
On Webmasters Stack Exchange, the following list of browsers supporting SNI is given: (accurate to November 2014)
Please also read this other question on Webmasters SE for more information about secure connections on shared and unique IPs.
Internet Explorer 7 or newer, on Windows Vista or newer.
Mozilla Firefox 2.0 or later
Opera 8.0 or newer (TLS 1.1 protocol needs to be enabled)
Opera Mobile with at least version 10.1 bèta on Android
Google Chrome (Vista or newer. XP on Chrome 6 or newer)
OS X 10.5.7 or newer on Chrome 5.0.342.1 or newer
Safari 2.1 or later (Mac OS X 10.5.6 or newer and Windows Vista or newer)
Konqueror/KDE 4.7 or newer
MobileSafari in Apple iOS 4.0 or newer
Android default browser on Honeycomb or newer
Windows Phone 7
MicroB on Maemo
The following browsers do not support SNI:
Internet Explorer, all versions, Windows XP
Safari, Windows XP
Windows Mobile 6.5 and lower
Android 2.x default browser