A detailed discussion of self-similarities in fragment-size distributions and fluctuations is presented using an exactly solvable model of fragmentation (the “chain model”). The effects of particle-number conservation and quantum symmetry can rigorously be considered in systems ranging from microscopic to macroscopic. Due to the analyticity of the model the various scalings can be studied free of any statistical noise. Using a tuning parameter we can generate self-similar distributions with realistic power-laws and/or fluctuations which show intermittency. Finite-size effects neither destroy nor cause intermittency. The relation of self-similarity in both the averages and the fluctuations can be studied analytically. It is found that they are unlinked - there are cases where the size-distribution is a power-law with realistic exponents τ between −2 and −3 but no intermittency. Two cases will even be shown which have indistinguishable fragment distributions but very different factorial moments. We also discuss the interpretation of both the size and slope of the factorial moments in terms of multiplicity and bin mixing. We show that while either is sufficient to produce large moments, one must have bin mixing to produce large slopes. The two types of mixing are necessarily linked in constrained systems such as described by our model.