Consider a $K$-user flat fading MIMO interference channel where the $k$th transmitter (or receiver) is equipped with $M_k$ (respectively $N_k$) antennas. If an exponential (in $K$) number of generic channel extensions are used either across time or frequency, Cadambe and Jafar showed that the total achievable degrees of freedom (DoF) can be maximized via interference alignment, resulting in a total DoF that grows linearly with $K$ even if $M_k$ and $N_k$ are bounded. In this work we consider the case where no channel extension is allowed, and establish a general condition that must be satisfied by any degrees of freedom tuple $(d_1,d_2,ldots,d_K)$ achievable through linear interference alignment. For a symmetric system with $M_k=M,$$N_k=N,$$d_k=d$ for all $k$, this condition implies that the total achievable DoF cannot grow linearly with $K$, and is in fact no more than $K(M+N)/(K+1)$. We also show that this bound is tight when the n-
umber of antennas at each transceiver is divisible by $d$, the number of data streams per user.
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