Any idea why we don't use significant digits in statistics? Something along the line of we are using estimates so rules about precision don't apply ;) ?
Significant digits are used in some fields (I learned about them in Chemistry) to indicate the degree of meaningful precision that exists in a number. This is an important topic in statistics as well, so in fact we report this constantly--we just report it in a different form. Specifically, we report confidence intervals, which indicate the level of precision of an estimate (such as a mean).
Once you've listed the 95% CI for an estimate, such as $(-0.12, 1.12)$, you can list as many digits for your mean as you might like, such as $0.50129519823975923$, and there is no problem. In fact, the statistician Andrew Gelman has recommended that you list at least four (2009, p. 4).
One reason for restricting the number of digits reported in many estimates, p-values, etc. is based on perception. Reporting something like p = 0.04872429 implies a level of precision in the results that causes them to be perceived as more accurate.
Essentially, the use of high numbers of digits in reporting statistical results tastes too many of trying to cloak your findings in an undeserved air of authority.
I think it really depends upon the level of confidence required, fewer digits for significance are appropriate for 95%, as opposed to 99.999% or greater, for example, as used by CERN for many of their results.
Are you talking about rounding your data to some number of significant digits or rounding your final answer? If you round your data you can get into situations where you've thrown away noise that statistical calculations need to use.