The universe appears to have a lower bound in the time dimension, why not an upper bound? The Big Bang looks like a lower bound to the "size" of the universe in the time dimension. Could it also have an upper bound, some furthest point in time from the Big Bang?
 A: It's certainly possible, though on current evidence it looks unlikely.
The past bound isn't really a bound in the usual sense of the word, but instead it's a singularity. If we solve Einstein's equations for the universe with a few apparently plausible assumptions we find that the universe is described by a scale factor, normally written as $a(t)$, and as this notation suggests the scale factor is a function of time, $t$. If we take any two points in the universe currently separated by some distance $d_0$ then the distance between those points varies with time as:
$$ d = a(t)d_0 \tag{1} $$
As the universe ages $a(t)$ gets bigger and $d$ increases, and this is why the universe is expanding. If we wind time backwards towards the Big Bang then $a(t)$ decreases and the universe contracts.
The problem is that as $t \rightarrow 0$ then $a(t) \rightarrow 0$, and therefore from equation (1) we find $d \rightarrow 0$. This means at time zero the spacing between every point in the (possibly infinite) universe was zero. As a side effect the density of the universe goes to $\infty$. This point is the Big Bang.
The Big Bang is singular because at that point we cannot use Einstein's equations to tells us what happened before it, so the singularity places a bound on our ability to calculate the behaviour of the universe. In principle time could extend backwards before the Big Bang to negative values, but we cannot calculate anything about the behaviour of the universe at those negative times.
As an aside, few physicists believe there really was a singularity at the Big Bang. Most of us believe that some form of quantum gravity becomes important at very high densities and this will prevent the density becoming infinite. For example Loop Quantum Cosmology predicts there was a Big Bounce. This is all wildly speculative, but if something like this did happen it means there was no past singularity and time extends smoothly backwards to $-\infty$.
But back to your question.
The point of all the above waffling was that the past boundary (if it exists) is due to a singularity, and likewise if there is a future boundary it too must be due to a singularity. In the early days of general relativity it was widely believed that the universe was closed and would recollapse in a Big Crunch. This would be a future singularity and would represent a future boundary of the sort you describe.
However it looks as if the universe is flat and won't recollapse so there is no Big Crunch to put an end to our timekeeping. About the only even remotely possible future singularity would be if dark energy has a particularly pathological equation of state, in which case there could be a Big Rip. This is a singular point and would create a future boundary. However you should appreciate that while the Big Rip is a fun idea there is absolutely no evidence to suggest it's likely to happen.
So the answer to your question is that no, there is (almost certainly) no future boundary to time.
