First of all in mathematical physics we are treating idealisations of the world out there. There has been a continuous debate from antiquity as to what is meant by a continuum - is it punctual, that is made from points, or is there more? In classical geometry as done by Euclid, he allows for points on lines but is silent as to whether lines are exactly all the points on the line. His geometry is synthetic. Descarte actually took that step (which paved the way for Calculus), his geometry is analytic.
Certainly the most common notion treats the continuum as made up of points, for example the calculus as envisaged by Newton & Leibniz does this. But already then Leibniz declared the neccesity for a analysis situs.
This was eventually formalised as topology. Over and above the idealisation of the continuum as a set of points one explicitly says how they cohere. It was also noticed that then one could actually do away with the notion of points altogether and just have the notion of cohesiveness (the theory of locales).
When you have a theory that has no points then in a precise sense it is meaningless to ask for exact simultaneity. But what this actually means is that one must expand what one can mean by this in this new context.
Whereas there has been a motion in Physics to atomising nature, from the atoms of matter, to quanta of energy and conjecturally spacetime; there has been an opposite motion in which the these atoms as point particles are seen to be problematic. Hence wave-particle duality (aka wavicles!), string and brane theory.
So far I've discussed simultaneity in the small which appears to be what you're asking about, but as a couple of the other posters have pointed out, there are problems with simultaneity in the large as elucidated by Einstein in his special theory. The laws of motion that we instinctively take for granted and formalised by Galileo are not correct - there is an absolute speed (the speed of light). Now speed as a formal concept ties together time and space, so how these are actually related must be modifed. This was Einsteins accomplishment. Whereas the older paradigm treated space & time essentially independent (so there are two notions of simultaneity - one for time & the other for space) he showed that they must be treated together (so there is a single notion of simultaneity using the idea of relativistic distance which combines information about both space and time).
It turns out we can keep the old idea of simultaneity when we are talking about events that are at the same location of space, but we cannot if they aren't, we must then use the relativistic notion.