The period of World War II would be a strong contender. Of course, it all depends upon one's definition of "independent countries", which can be highly contentious. For the purposes of this answer, I will use the definition outlined my the Montevideo Convention:
"The state as a person of international law should possess the following qualifications: (a) a permanent population; (b) a defined territory; (c) government; and (d) capacity to enter into relations with the other states."(1)
Using this criteria, it is clear that during WWII there were relatively few independent states compared to any other time period. This was primarily due to the Axis conquests that brought many formerly independent states under foreign rule. The nadir for the number of independent states was likely 1944, following the collapse of Italy's government, as well as Iran's division into spheres of influence by the USSR and the British Empire.
The online historical atlas Geacron(2) shows that there were 52 sovereign nations in the year 1944. This map depicts several central European states as independent, although they were heavily influenced by Nazi Germany, especially Slovakia, Croatia, and Hungary.(3)
There are a number of other eras that might be considered for this question. For instance, during the invention of agriculture, there were obviously very few states that functioned as organized governments. However, outside of the few agricultural societies, there were thousands of hunter-gatherer groups that were independent. A similar dynamic existed during the period of early European colonization, up until the Scramble for Africa- a small number of European empires appear on the map, yet there were countless non-European entities, both in the Americas and in Africa, that were not known or formally recognized.
Sources: 1)http://www.cfr.org/sovereignty/montevideo-convention-rights-duties-states/p15897
3)Axworthy, Mark W. (2002). Axis Slovakia: Hitler's Slavic Wedge, 1938-1945. Europa Books Inc. p. 45. ISBN 1891227416.
This is a pretty dicey question, due to the variable number of qualifications and the open-endedness of the question. "Country" usually is taken as "independent state," with the implication of being internationally recognized(so Transnistria, Rep. Srpska would be out. )
Although the question is a weak one, the logical point would be 1914 or 15: The various empires had gobbled up Asia, Africa and Oceania, and the Austrian, Russian and Ottoman empires were not yet broken up.