Is it possible for two events happen at the exact same time? Is it possible for any two events to occur at the exact same time?
As I see it, because time intervals can always be split up into smaller units (it is infinitely divisible), we can always be more and more exact with measuring the time at which something happened, until we find out that the two things did not in fact occur at the same time. Is this correct, or is it actually possible for two things to happen at the exact same time?
 A: 
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.
A: Events are points $(x,t)_S$ onto a chart $S$ on some space-time manifold and in this respect whenever two such points $P_1 = (x_1,t), P_2=(x_2,t)$ have the same $t$-coordinate in that reference frame then yes, they do occur at the same time for the observer described by the chart $S$. For another observer, represented by a different chart $S'$, the $t$-coordinate may transform and give two different times $P_1'=(x'_1,t'_1), P_2'=(x'_2,t'_2)$ with $t'_1\neq t'_2$; therefore, although the two events were simultaneous in the first reference frame, they are not in the second one.

As I see it, because time can always be split up into smaller units

does not really make much sense, because you do not split coordinates into smaller units. What you perhaps have in mind is splitting time intervals into smaller pieces, but that is a totally different thing. But even in that case, however you split your interval up, the total interval length (i. e. the integral) is always the same, no matter the procedure you use; hence you can exactly compare different intervals: if they are the same then they are the same, if not then they are not.
How to experimentally measure time intervals in order to reduce the uncertainties is anyway a totally different question.
A: 
Is it possible for two events happen at the exact same time?

No.
Even at any one event itself there can be several (or in though-experimental principle even arbitrarily many) distinct participants (encountering and passing each other, momentarily). All their individual distinct times (indications) are attributable to this one event of their meeting.
Consequently, one particular time cannot be attributed to two distinct events;
and the simultaneity relation by which to determine whether the time of one participant at one event corresponds (or, in an appropriately lose sense, is equal to) the time of a (suitable) other participant at another event cannot be extended into a consistent relation between two entire (separate) events.
Your subsequent argument is somewhat unrelated to your question itself (apart from quibbles about terminology):

we can always be more and more exact with measuring the time at which something happened, until we find out that the two things did not in fact occur at the same time.

Not quite:
If we develop means for improving the resolution by which to determine whether two observed indications (times) belonged to two distinct events, or to one and the same event, then this still includes the result value "same event" in the image of the measurement operator being applied. (Of course, dynamic characterizations such as "energy" or "momentum" of individual participants with respect to each other would be correspondingly less resolved, or even completely undefined.)
A: In a non-relativistic point of view, yes. Events occurring in less than $5.39106 × 10^{-44} s$ (the Planck time) are considered to be simultaneous. The Plank time is the time it takes light to travel $1.1616 × 10^{-35}m$, about $1/(2.67×10^{24})$ the size of the hydrogen atom. The Plank time is also, considered to be the smallest time lapse that could ever be measured, but this is under debate.
Actually, the smallest time interval uncertainty in direct measurements is about $3.7 × 10^{26}$ Planck times (wikipedia).
From a relativistic point of view, simultaneity depends totally on the observers frameworks, so, two events that seems simultaneous to one observer (Alice, just to name her), may not be simultaneous to another observer (Bob, if you like it) if they are on distinct frameworks.
