Can neurotransmitter concentrations in the synaptic cleft add up if there are several impulses in quick succession?
The answer by bpedit is a good explanation of temporal/spatial summation, which can be partly a postsynaptic phenomenon (due to membrane time constants or second messenger sytems, for example).

Your specific question is about whether extracellular neurotransmitter levels sum over time. This definitely occurs regularly with "neuromodulatory" neurotransmitters like serotonin (others that behave similarly include dopamine and norepinephrine/epinephrine). "Faster" neurotransmitters like glutamate and GABA are cleared more quickly, though there are complicating factors there as well that are outside the scope of this question.

Although the summation has been shown many times, I looked quickly for an example and found the paper by Bunin and Wightman, 1998. Figures 2-6 all show extracellular serotonin concentrations that are elevated for seconds after release (let me know in the comments if you can't access this paper at your institution), and they show the changes in extracellular concentrations with multiple stimulus pulses (see in particular Figure 3).