Can you keep your phone and internet and drop your cable tv?

There are two ways that I know of that you can get your own internet connection without a phone line or cable TV.

1) With a public WiFi connection in a hot spot.

These are available in various locations inclduing pubs, cafes and certain phone boxes operated by BT in the UK provide WiFi access to passing compatible devices, however the coverage is limited and it is very expensive compared to...

2) Connecting a mobile phone or mobile modem to your computer.

The majority of these require a fixed term contract. Indeed, in the UK only Three offer a Pay As You Go option. The mobile network is much more available although unless you have access to a 3G signal it can be very slow. The disadvantages of this are that a monthly contract is required and if you are paying a fixed rate for a certain amount of bandwidth followed by a more expensive per megabyte rate you need to watch what you're downloading. If you can find a Pay As You Go operator with a reasonable unlimited broadband rate and can find a USB modem SIM-free then it can be very cost-effective with a one-off payment with only the charges for the bandwidth you use at a much cheaper rate than public WiFi hotspots.

3) WiMax & Satellite

There are also other wireless solutions available that do not require a mobile phone. Clear.com offers limited Wimax coverage in several US cities and satellite internet providers can give you service in remote areas.

Other than that you maybe able to find public internet terminals where you are, but bear in mind that this is not your internet connection and will be very likely blocked or filtered, especially in a public library.

Also, while this information relates particularly to the UK as this is where the contributor is from, you may be able to find similar options in your country, check for WiFi coverage and with your local mobile operators.