This can't work or somebody would be doing this already.
Well, it could be done - either in a server, or in the internet radio device. But if it's done in the radio then it needs more memory.
By way of an example, here in the UK we have conventional radios for receiving terrestrial digital broadcasts (using the DAB system) which are able to buffer the broadcast mpeg data stream. When you press the pause button you get as much as an hour's buffering and you can move back and forth through the buffer to listen to from a particular point. While it's a nice idea, it is often not compelling enough for consumers to pay the additional cost. Since only the minority of radios have this feature they are significantly more expensive than those without this feature (it not just the cost of the RAM, its the usual mark-ups in the supply chain that do the damage).
In your example, if done in a server, memory capacity would not be an issue. It just needs someone to write it. (You can tell I'm a hardware guy - I always trivialise this software stuff...)