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Stars that pass too close to a super-massive black hole may be disrupted by strong tidal forces. OGLE16aaa is one such tidal disruption event (TDE) which rapidly brightened and peaked in the optical/UV bands in early 2016 and subsequently decayed over the rest of the year. OGLE16aaa was detected in an XMM-Newton X-ray observation on June 9, 2016 with a flux slightly below the Swift/XRT upper limits obtained during the optical light curve peak. Between June 16-21, 2016, Swift/XRT also detected OGLE16aaa and based on the stacked spectrum, we could infer that the X-ray luminosity had jumped up by more than a factor of ten in just one week. No brightening signal was seen in the simultaneous optical/UV data to cause the X-ray luminosity to exceed the optical/UV one. A further XMM-Newton observation on November 30, 2016 showed that almost a year after the optical/UV peak, the X-ray emission was still at an elevated level, while the optical/UV flux decay had already leveled off to values comparable to those of the host galaxy. In all X-ray observations, the spectra were nicely modeled with a 50-70 eV thermal component with no intrinsic absorption, with a weak X-ray tail seen only in the November 30 XMM-Newton observation. The late-time X-ray behavior of OGLE16aaa strongly resembles the tidal disruption events ASASSN-15oi and AT2019azh. We were able to pinpoint the time delay between the initial optical TDE onset and the X-ray brightening to $182 \pm 5$ days, which may possibly represent the timescale between the initial circularization of the disrupted star around the super-massive black hole and the subsequent delayed accretion. Alternatively, the delayed X-ray brightening could be related to a rapid clearing of a thick envelope that covers the central X-ray engine during the first six months.