Pre- and post-measurements have been taken in 3 different conditions, and I was wondering if a paired t-test would be an appropriate statistical analysis to use?
This is a 2 (time: pre vs. post) x 3 (condition: a vs. b vs. c) design, where the former factor is repeated-measures (also called within-subjects) and the latter is between-subjects (that is, people are only in one of those conditions).
Probably, what you want to know is, "Is the post measurement better than the pre measurement?" and "Does one of the conditions perform better than the rest?"
What you can do is a 2 x 3 mixed design analysis of variance (ANOVA). You can see if there is a significant interaction between time and condition—this could show you if certain conditions performed better than others. You could also look at just the main effect of time to see if all conditions combined performed better. Note that, if you ignore condition, you could do a paired t-test between pre vs. post; this would not tell you any information about the condition, though, which you probably want to know about. There are a lot of introductory statistics books and introductions online about mixed ANOVAs.
Additionally, you could do a multilevel model (also called hierarchical models or mixed effects models). You would specify measurement as Level 1, with each person having 2 observations. You would then have a Level 1 predictor of pre or post. These Level 1 observations would be nested within person. And the predictor at Level 2 would be what condition they were in. Then you could also specify an interaction between time and condition. You would also want to specify a random slope for time at Level 1. There are also a bunch of books and tutorials on these multilevel/hierarchical/mixed models, as well.