If we send light down an optic fibre and loop the fibre back onto itself, with just an opening for our light source, would the light (photons) accumulate within the fibre? If so then to what extent would they accumulate and can it be calculated? Would this be the bandwidth of the fibre?
Yes, you can create a 'ring' with optical fiber, with a coupling section to add photons in. You can keep adding photons until the (non-linear) loss mechanisms balance the input. This can be estimated, but is hard to calculate from first principles because you don't know all the parameters for the actual assembled setup. This has nothing in particular to do with the bandwidth of the fiber, although where your laser is with respect to the loss profile vs wavelength impacts the loss mechanisms.
Yes, this is called a fiber ring resonator, and has been studied long before the fiber was even invented, using planar lightwave circuits for instance. The physics is pretty similar to a Fabry Perot resonator, in that you have longitudinal resonant modes which correspond to constructive interference of the light on each round-trip. With low propagation losses, it is possible to achieve field enhancement factors of several hundreds when the resonant condition is achieved.