Berlin

For a list of special events in Berlin by the month, take a look at the website Berlin.de

late June. European school children are all off at some point between early July and late August. Dodge the worst of the crowds by skirting around these periods. If you're up for a major public holiday event, the day of German unity ( Tag der Deutschen Einheit in German) is the anniversary of the Berlin wall falling and as important to Germans as Independence day is to the Yanks.

A month in Berlin's a bit hardcore though - but you could do a grand tour of Germany in that time.

Dunno that OP means a full month - I think just "during which month".

June, July or August would be the best time, whatever you choice.

Late June / mid July. Early June can still be a bit wet.

August has lovely weather for lying about the parks but it feels just so wrong to spend August in a city!

August this year regularly hit temperatures in the high 30s - not very pleasant for sightseeing (brilliant for lazing around at the lakes though!)

If you are culture vultures then I think earlier in the summer or early autumn would be best. September this year was glorious, for example - and the kids are back at school.

I was in Berlin in August a couple of years ago -daytime temps in the mid 30s and wind straight from the Sahara, so was pleased to leave. Personally think September ideal, otherwise late June or early July. But like #6 says, leave time to go to the lakes - not just Wannsee, but look at going to Strausberg, Woltersforf or the Müggelsee beyond Köpenick

if you are asking which month is the best weather wise, there is no way anybody can answer that. normally july and august would be the warmest month, but there is no security on that. this summer (2015) was great, last summer was horrible. so its impossible to tell you when you should spend your time in berlin. do go when you want to go and make the best of it.

what sort of culture? the classic venues like opera/concert etc. usually run until late June/early July before their summer break and usually won't return to action before September. all sorts of festivals during summer, though, so make up your mind.
if there are museums you're specially interested in, consider checking for special exhibitions, maybe there are some that end/start before/in/after summer (it's still a bit early for checking, though).
Berlin is always busy, yet when the school year draws to its end, the city will be crawling with classes on field trips - I'd prefer school holiday time itself, more relaxed imo.