We present the results of Chandra X-ray Observatory (CXO) observations of the planetary nebulae (PNs) NGC 40 and Hen 2-99. Both PNs feature late-type Wolf-Rayet central stars that are presently driving fast (∼ 1000 km s −1 ), massive winds into denser, slow-moving (∼10 km s −1 ) material ejected during recently terminated asymptotic giant branch (AGB) evolutionary phases. Hence, these observations provide key tests of models of wind-wind interactions in PNs. In NGC 40, we detect faint, diffuse X-ray emission distributed within a partial annulus that lies nested within a ∼ 40 ′′ diameter ring of nebulosity observed in optical and near-infrared images. Hen 2-99 is undetected. The inferred X-ray temperature (T X ∼ 10 6 K) and luminosity (L X ∼ 2 × 10 30 ergs s −1 ) of NGC 40 are the lowest measured thus far for any PN displaying diffuse X-ray emission. These results, combined with the ring-like morphology of the X-ray emission -2 -from NGC 40, suggest that its X-ray emission arises from a "hot bubble" that is highly evolved and is generated by a shocked, quasi-spherical fast wind from the central star, as opposed to AGB or post-AGB jet activity. In constrast, the lack of detectable X-ray emission from Hen 2-99 suggests that this PN has yet to enter a phase of strong wind-wind shocks.