Fibers themselves have an extremely high bandwidth

There is some very recent work on the use of fibres with few (i.e. fewer than about five) bound eigenfields and the encoding of separate, potentially petabit per second each, channels, one for each bound eigenfield. See the work of Love and Riesen, e.g. Optics Letters 37, 19 (2012) 3990-3992.

Fibers themselves have an extremely high bandwidth in principle. Pretty much all the wavelengths where they are transparent enough to transmit light such that you can still detect it at the other end.

Where the fiber itself is the limiting factor is dispersion, ie. since all signals have a bandwidth themselves, their 'red' and 'blue' portions travel at different speeds. So if the fiber is long enough and your signal modulation is very fast, at the detector end the square input pulse will be rounded enough that you have trouble distinguishing it from the previous or following one.
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The strongest limits on the usable bandwidth come from the lasers and detectors that are being used. To get all the different channels in and out while keeping them separate, you need lots of narrow band filters and modulators/demodulators. That part of the technology is expensive, but is more often replaced/upgraded than the fiber itself. The above is mostly relevant for long-haul fibers.