Qualifier: I am not a statistician. I have 4 data sets and want to determine whether the 4 regression coefficients obtained are statistically significantly different. I compared each pair of coefficients separately (6 hypotheses) with t-tests and found no significant difference (t values ~1 for 6 d.f.). Does this support my null hypothesis (there is no significant difference in the coefficients) or do I need to make a correction because I am comparing multiple hypotheses. That is, if I made a correction would it only make the level of significance even lower?
Answering briefly on this.
Using a P-value in Null Hypothesis Significance Testing (NHST) does not allow you to draw conclusions about your null hypothesis, just that you cannot reject it at the observed data. This is a common misconception. See e.g. Nickerson, R. S. (2000). Null hypothesis significance testing: A review of an old and continuing controversy. Psychological Methods, 5, 241-301.
Instead, you could calculate confidence intervals to get an idea about the population estimate, see for example Abserson 2002 - or use Bayesian statistics, but that is another chapter.
You are right that your P-values will be even higher when correction for multiple comparisons is performed.