Leo T is the lowest mass gas-rich galaxy currently known and studies of its gas content help us understand how such marginal galaxies survive and form stars. We present deep neutral hydrogen (H I) observations from the Westerbork Synthesis Radio Telescope in order to understand its H I distribution and potential for star formation. We find a larger H I line flux than the previously accepted value, resulting in a 50% larger H I mass of 4.1 x 10(5) M-circle dot. The additional H I flux is from low surface brightness emission that was previously missed; with careful masking this emission can be recovered even in shallower data. We perform a Gaussian spectral decomposition to find a cool neutral medium component (CNM) with a mass of 3.7 x 10(4) M-circle dot, or almost 10% of the total H I mass. Leo T has no H I emission extending from the main H I body, but there is evidence of interaction with the Milky Way circumgalactic medium in both a potential truncation of the H I body and the offset of the peak H I distribution from the optical center. The CNM component of Leo T is large when compared to other dwarf galaxies, even though Leo T is not currently forming stars and has a lower star formation efficiency than other gas-rich dwarf galaxies. However, the H I column density associated with the CNM component in Leo T is low. One possible explanation is the large CNM component is not related to star formation potential but rather a recent, transient phenomenon related to the interaction of Leo T with the Milky Way circumgalactic medium.