Cable television programming is typically delivered through a cable-company-provided device referred to as a "set-top" box. In theory it is legally and technically possible to for the consumer to supply their own set-top equipment to consume cable service, however as both technology and policy have been evolving this is a murky area.
Cable service providers have been steadily moving all content (channels) to encrypted signals ("encrypted QAM"). In order to access this content (with your own hardware) it is necessary to have a "CableCard"-compatible tuner and cable-company provided CableCard (which you rent for a separate fee) which acts as a special hardware key and decoder for this content. There are a handful of products on the market, including TiVO and consumer PC components for DIY "Home Theater PC" systems (specifically Windows Media Center).
See for background: http://www.usatoday.com/story/tech/columnist/2013/04/14/cable-encryption/2081471/
If you can provide your own tuning equipment you may able to avoid Time Warner's set-top box leasing fees through the "Bring Your Own Box" discount: http://www.timewarnercable.com/en/residential-home/support/topics/tv/cablecard/byob-discount-form.html
Time Warner currently provides a Roku app and has given some indications that it may be opening up third party device access in the future: http://www.cnet.com/news/time-warner-cable-will-let-you-junk-your-set-top-box-next-year
If you are interested in exploring the possibility of using your own cable service tuning equipment, the following resources may be helpful:
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