How the size of organelles in cells is controlled despite a constant turnover of their constituent parts is a central problem in cell biology. A general mechanism has been proposed based on the idea that an organelle grows by self-assembly of molecular subunits that freely diffuse in the cytoplasm. Assembly continues until the available pool of subunits is depleted to the point when the stochastic addition and removal of subunits is balanced, leading to a structure of well-defined size. Here we focus on length control of multiple filamentous structures in cells, such as actin cables and flagella. Using queueing theory and computation we show that the limiting pool mechanism leads to three different phases of assembly, starting with a rapid growth phase when all filaments quickly accumulate a large number of available subunits. Then, the slower growing filamentous structures enter a disassembly phase as they gradually lose all of their subunits to the faster growing structures. Finally, when multiple, equivalent fast-growing filaments are present, their lengths undergo protracted diffusive dynamics due to the stochastic swapping of subunits between them. This eventually leads to a broad, power-law distribution of filament lengths in steady state. Our findings demonstrate that the limiting-pool mechanism is incapable of controlling lengths of multiple filamentous structures that are assembled from a common pool of subunits, and at best, can produce only one filament of a well-defined size. Overall, our theoretical results reveal physical limitations of the limiting-pool mechanism of organelle size control.