Dimension counts for limit linear series on curves not of compact type

We first prove a generalized Brill-Noether theorem for linear series with prescribed multivanishing sequences on smooth curves. We then apply this theorem to prove that spaces of limit linear series have the expected dimension for a certain class of curves not of compact type, whenever the gluing conditions in the definition of limit linear series impose the maximal codimension. Finally, we investigate these gluing conditions in specific families of curves, showing expected dimension in several cases, each with different behavior. One of these families sheds new light on the work of Cools, Draisma, Payne and Robeva in tropical Brill-Noether theory.