Not sure i would consider 5 weeks a small part, after all if you are traveling to Japan in that time frame it could be 100% of your summer in Japan, but yes, 2/3 or so of summer that isn't the "rainy" season but it's hotter in those months and there is still a lot of rain, it's humid and hot in summers in Tokyo. Looking at the averages, there is usually more rain in August and even September than July.

Is anyone really trying to deny the fact that Tokyo is miserable in the summer in terms of heat and humidity? Especially since AC is used a lot less commonly and usually set at higher temperatures than what what is used in the states... I'd say it's a pretty safe bet that a traveler from the USA, even somebody from the south or Florida, would not find the heat and humidity nice during summers in Tokyo!