Today, hydrologic research and modeling depends largely on climatological inputs, whose physical and statistical behavior are the subject of many debates in the scientific community. A relevant ongoing discussion is focused on long-term persistence (LTP), a natural behavior identified in several studies of instrumental and proxy hydroclimatic time series, which, nevertheless, is neglected in some climatological studies. LTP may reflect a long-term variability of several factors and thus can support a more complete physical understanding and uncertainty characterization of climate. The implications of LTP in hydroclimatic research, especially in statistical questions and problems, may be substantial but appear to be not fully understood or recognized. To offer insights on these implications, we demonstrate by using analytical methods that the characteristics of temperature series, which appear to be compatible with the LTP hypothesis, imply a dramatic increase of uncertainty in statistical estimation and reduction of significance in statistical testing, in comparison with classical statistics. Therefore we maintain that statistical analysis in hydroclimatic research should be revisited in order not to derive misleading results and simultaneously that merely statistical arguments do not suffice to verify or falsify the LTP (or another) climatic hypothesis.