I conducted a paired samples t-test to analyse the effect of an exercise intervention on blood pressure measurements. The test demonstrated a significant p-value. However, when I have calculated 95% confidence intervals for the mean difference between pre and post measures the result suggested there to be no significant difference between pre and post measures, i.e lower level minus and upper level positive. Is this possible?
A 95% CI and a test done with an alpha of .05 are completely equivalent. There must be a problem either with your test or your calculation of the CI. As gung suggests, this is more likely a result of how you calculated your CI than a problem with the test.
Based on your descriptions in the comments, you found the right effect size, but your error term is wrong as is your critical value. A paired samples t-test is equivalent to a one-sample t-test of the differences between group (against H0 that the mean difference score is 0). So the error term you want is the standard error of the mean for a one-sample t-test (if memory serves, the STD of the difference scores over the square root of n). You also want to use the critical value for a 9 df t-test, 2.262157, I believe, not 1.96.