The Divergence Problem refers to an anomalous reduction in forest growth indices and temperature sensitivity which has been detected in tree-ring width and density records since around the middle 20th century in forests from the Northern Hemisphere. In essence, while instrument temperature readings have shown distinct temperature increases since that time, tree ring records, which are often used as "proxies" to determine temperature changes in the period before instrumentation, have not shown a comparable increase.
Obviously, if the trees sampled in tree ring record studies differ over time in their sensitivity to temperature change, this lessens their ability to serve as proxies, and makes it difficult, for example, to come to any conclusion as regards to modern temperatures vs. those experienced during the alleged "Medieval Warm Period".
The issue has been examined extensively since it was identified in the mid-1990s, and a number of studies have suggested that the divergence witnessed is a recent, one-off event brought about by the effects of "anthropogenic forcings" on the forests where it has been seen. This conclusion has been dismissed by climate changed skeptics as invoking warming to explain why there is no evidence of warming in recent tree-ring cores. However, to understand why it still serves as at least a good partial explanation for the Divergence Problem, take a look at this very recent (not yet published, as far as I can tell) paper by Rosanne D’Arrigo et al, which summarizes a decade of literature on the topic. While concluding that there is probably not "a single "divergence" phenomenon", Arrigo also points to several consequences of global warming that may well impede the ability of Northern Hemisphere forests to act as natural thermometers and, more ominously, could slow or reverse their future ability to sequester carbon.