I need to test some before and after data, to see if there is a significant shift (n = 114) but the data differences are not normally distributed (or, at least, a histogram would seem to indicate that). Can I use a paired t-test?
Have you thought about a non-parametric test such as the Mann-Wilcoxon? You could test the before and after distribution to see if the average value has shifted significantly in one direction or the other.
@Scortchi is correct in that in your specific case, the Wilcoxon signed rank test would be a more appropriate non-parametric test.