We often hear that it's important to learn from history to avoid the errors of people before us. To me - then - it's astounding to think that a head of state like Hitler, surrounded with elite military strategists, was persuaded to attack Russia in the wrong season, duplicating the error made by Napoleon I.
There is no good season for invasion of Russia
Napoleon and Hitler invaded in the beginning of the summer. Weather in this part of the year is relatively good for advance, and Russian roads are passable. Napoleon went straight to Moscow and captured it relatively early in the year ( September 14, 1812). However, he did not capture Kiev and his army could not move against St Petersburg. Consequently, his deep and thin penetration into Russia depended on a single route (through Smolensk) which was constantly harassed. Napoleon was compelled to withdraw, first through muddy autumn weather, and then through early Russian winter (November and early December) which ended the way it ended.
Hitler was more cautious in this regard - he moved to capture Kiev first, and besieged Leningrad. Only after that was he able to go to Moscow, but at that time the autumn rasputitsa settled in, and this the slowed German advance until ground froze in November. Unfortunately for the Germans, they were not prepared for a winter campaign and again the rest is history.
Before French and German invasions, it is worth to mention Swedish invasion of Russia which actually did start in the middle of the winter. However, despite being better prepared for winter warfare then later French and Germans attempts, Swedes were also forced to remain in winter quarters after initial trust. Consequently, they only resumed campaign in July, after spring rasputitsa has passed, and after that their attempt met similar fate as their successors, with many troops lost during Great Frost of 1709.
Overall, conventional wisdom would be to wait for spring mud season to end, this usually happens in late May. Do to various circumstances, invaders usually cannot attack right away, forcing the beginning of operations to late June instead. After that it is race against time until October rains start and again create autumn rasputitsa followed by dreaded Russian winter . This actually leaves barely four months of good campaigning weather, which definitely is not enough to defeat Russia.