How is it possible that we haven't discovered anything in the Oort cloud yet? The Oort cloud is a hypothetical cloud of small icy bodies surrounding the Sun at more than 1000 AU. It is thought to be a vast reservoir of comets that occasionally get disrupted, sending comets towards the inner solar system.
I would guess that there are stellar occultations where a body from the Oort cloud passes in front of a background star, obscuring its light. This makes it surprising to me that we haven't detected any objects in the Oort cloud. Perhaps the sensitivity of our instruments is not high enough? Perhaps that in spite of the large number of objects that should exist in the Oort cloud, these occultations are actually infrequent, and you would have to wait tens of years before seeing one? Perhaps nobody has combed the data to find such signals?
Would any of our existing telescopes have been able to pick up the occultation if it was pointed in the right direction at the right time?
It's not very surprising that no Oort Cloud objects have been detected via occultations. They're extremely rare, even for our most advanced space-based observatories.
According to Ofek &amp; Nakar 2010, published about one year after the launch of Kepler, the telescope would be able to detect between $0$ and $\sim100$ occultation events of Oort Cloud objects, assuming that it monitored $\sim10^5$ stars for a time $\tau=3$ years - both of which ended up being underestimates.$^{\dagger}$ Part of the reason for the uncertainty is that the parameters of the Oort Cloud are not tightly constrained. Varying the inner radius $r_{\text{min}}$ from 1000 to 5000 AU, or varying the index $\alpha $ of the number density distribution $n\propto r^{\alpha}$ from $-4$ to $-3$, can produce changes on the expected occultations by a couple orders of magnitude.
The problem becomes apparent when you consider that 1) occultations are intrinsically short and infrequent and 2) they're one-off events. It's not the same as detecting a transiting exoplanet, or observing an expected occultation from a minor planet whose orbit is already known. Therefore, it's hard to tell whether an event is a true occultation or simply a glitch of some sort. The second point of the paper was to present methods for validating possible events.
Ground-based surveys of Kuiper Belt Objects (KBOs) have been performed, but they suffer from atmospheric effects, as you might expect. Given that occultations would last on the order of 1 second, atmospheric scintillation becomes a problem for the high-cadence searches required (Alcock et al.). Given that some ground-based surveys just for KBOs a few kilometers across, like the Taiwanese-American Occultation Survey, have returned no detections (Zhang et al. 2008), it might not be surprising that ground-based Oort Cloud searches are also extremely difficult.

${\dagger}$ By comparison, the CoRoT mission, assuming it monitored $\sim10^4$ stars over the same period, was expected to detect essentially no occultations, according to the same analysis.