I am using Time Warner Cable's most popular provided box...PIONEER. It looks identical to this one:

I emailed TWC to asked about owning your own equipment, and this what they reply with:

"Thank you for your most recent message to Time Warner Cable of NYC.

You are correct that, under the Cable Act, "cable operators may not prevent
customers from using their own equipment if such equipment is technically compatible
with the cable system."

As you may be aware, under rules recently adopted by the FCC, cable operators are
required to provide separate security components to customers who purchase
compatible converters from third party retailers, to enable use of the converter.
While the separate security components are available from Time Warner Cable, it is
our understanding that compatible converters are not available at this time from
third-party retailers. Once these compatible converters have been made commercially
available at retail, if you elect to purchase one, we will be happy to provide you
with the necessary device."