A number of different proteins possess the ability to polymerize into filamentous structures. Certain classes of such assemblies can have key functional roles in the cell, such as providing the structural basis for the cytoskeleton in the case of actin and tubulin, while others are implicated in the development of many pathological conditions, including Alzheimer's and Parkinson's diseases. In general, the fragmentation of such structures changes the total number of filament ends, which act as growth sites, and hence is a key feature of the dynamics of filamentous growth phenomena. In this paper, we present an analytical study of the master equation of breakable filament assembly and derive closed-form expressions for the time evolution of the filament length distribution for both open and closed systems with infinite and finite monomer supply, respectively. We use this theoretical framework to analyse experimental data for length distributions of insulin amyloid fibrils and show that our theory allows insights into the microscopic mechanisms of biofilament assembly to be obtained beyond those available from the conventional analysis of filament mass only.