The light intensity of a LED is not directly following the current, but there can be additional delay for a build-up / decay of excess minority carriers. Especially some IR and red LEDs can be quite slow. One could get an idea on how much delay by looking at the reverse recovery of the LED.
It's my understanding that LEDs peter out above a few 10s MHz, for this reason.
There have been proposals for highly-asymmetric "LiFi" using illumination LEDs (basically bathe the room in data, let the clients sort it out; upload has to be handled separately, perhaps with a very low BW IR link), which would be "competitive" with WiFi, so, about this bandwidth (that is, the 2.45GHz WiFi, with many users and mediocre signal quality, giving 10s Mbps per client).
Incidentally, notice the yellow phosphors have longer delay times, so this would require blue filtered photodiodes to receive such data.
Far higher bandwidths are possible, optically, of course -- but they require laser diodes and/or acousto-optical modulators. The power levels are quite modest, suitable for fiber use, nothing you'd be able to broadcast (well, "broad"cast; with good optics, a cableless point-to-point link might do).
Tim