Modal Team Logic (MTL) extends Väänänen's Modal Dependence Logic (MDL) by Boolean negation. Its satisfiability problem is decidable, but the exact complexity is not yet understood very well. We investigate a model-theoretical approach and generalize the successful filtration technique to work in team semantics. We identify an "existential" fragment of MTL that enjoys the exponential model property and is therefore, like Propositional Team Logic (PTL), complete for the class AEXP(poly). Moreover, superexponential filtration lower bounds for different fragments of MTL are proven, up to the full logic having no filtration for any elementary size bound. As a corollary, superexponential gaps of succinctness between MTL fragments of equal expressive power are shown.