After 2000, Montenegro has been hit by several severe droughts. The results presented in this study were based on the analysis of the standardized precipitation index (SPI) and the standardized precipitation evapotranspiration index (SPEI) for the period 1961–2020. They showed the prevalence of a negative trend in both indices and a significantly higher frequency of drought in the second half of the observed period (1991–2020). The SPEI index is a more representative indicator of drought, because in addition to precipitation, it also takes into account potential evapotranspiration. Using the rescaled adjusted partial sums (RAPS) method, it was determined that the tipping point when the SPEI suddenly fell was the mid-1980s. Out of the 15 analyzed teleconnection variation indicators (atmospheric and oceanic oscillations), 11 oscillations have a significant impact on one or more of the 24 analyzed SPEI time series. The consequences of the drought in Montenegro, as part of the Mediterranean, are most noticeable in the summer, primarily in the form of water shortages, dried vegetation, and frequent fires. Decision-makers in Montenegro should pay attention to this extreme, as drought could pose a serious problem under the projected warmer climate conditions.