  Some new necessary conditions for the existence of vector space partitions
are derived. They are applied to the problem of finding the maximum number of
spaces of dimension t in a vector space partition of V(2t,q) that contains m_d
spaces of dimension d, where t/2<d<t, and also spaces of other dimensions. It
is also discussed how this problem is related to maximal partial t-spreads in
V(2t,q). We also give a lower bound for the number of spaces in a vector space
partition and verify that this bound is tight.
