Is it possible for any two thing to occur at the exact same time
This is a physics question and answer site. In physics our examination of nature has shown that there exist many frameworks for defining "events" , as in your title, or "things" as in your question.
The main frameworks where "simultaneity" and "event" have to be defined so as to make the question clear are: classical mechanics, electrodynamics, thermodynamics, classical statistical mechanics, and quantum mechanics with its quantum statistical mechanics. These frameworks have a region of validity in (x,y,z,t) and numerous other prerequisites ( General relativity? special relativity? underlying coordinates) to be defined clearly, but they join smoothly (mathematically) into each other as the range of validity changes from one framework to another.
Look at this table of times , fluorescence is a quantum mechanical effect and thus a nanosecond can be seen as the interface between classical and quantum mechanics.
In classical mechanics the range of validity in space for an event to be defined, lets say a classical particle (billiard ball like) hitting another one: it ranges from a micrometer ( below this the quantum mechanical frame starts to be important) and times of nanoseconds ( below this one is in the quantum mechanical framework).In this framework two events will be simulataneous within these a micrometer and a nanosecond. This will be the exact same time in classical mechanics.
In quantum mechanics life is not so simple, there exist probabilities in defining positions and times, controlled by quantum mechanical solutions of boundary value problems. The Heisenberg uncertainty principle contains the probabilistic uncertainties observed in quantum mechanical events.
When the particles in the event are in the QM framework, a proton hitting a proton for example, there is no unique number for its location, it will depend on the energy they carry and there will exist an uncertainty within a box of delta(E)*delta(t) of where exactly the particles will appear at a specific event/measurement. One cannot have a fixed definition of simultaneity at the interactions of quantum mechanical entities, it will depend on the particular event. Very small times have been measure as seen in the table linked above, at the expense of the energy to keep the HUP satisfied.
In this report ,
Researchers used short pulses of laser light to produce images of electrons leaving atoms and recorded what happened to within 100 attoseconds.
So this seems to be the smallest quantum mechanical limit of separating in time events for this particular experiment.
In conclusion , the "exact same time" event though it can be defined in the classical framework, it has the limits of the existence of the quantum mechanical framework in its definition of "exact same time", and in the quantum mechanical framework the answer is no, because the HUP introduces an uncertainty in time and energy that has to be obeyed and will introduce an uncertainty in simultaneity depending on the experiment.