Madrid in May? June?

https://www.timeanddate.com/weather/

A source for weather histories

There are several websites where you can find historical temperature data per day. This is more relevant than 'averages'. I prefer Wunderground for this information because it also includes precipitation; but there are others. This link should take you to the calendar view for Madrid in May 2017 https://www.wunderground.com/calendar/es/madrid/LEMD/date/2017-5
I tend to check the last five years for the time period I may be somewhere and check if there are trends. I was surprised how warm it could be in Madrid then.

I'd also check actual, historical, day-by-day weather stats for the Dordogne. Use one of the interior towns if that's where you'll be, rather than Bordeaux, which might be different due to proximity to the coast. The Dordogne is likely to be cooler and wetter than Madrid, so it will be a balancing act.

Madrid is basically high plains, so it can get deathly hot in late summer and warmer early on, but the humidity stays a bit lower than the coast. May/June should be comfortable, not too wet, but as you experienced in Italy, if you are there only a few days, then anything can happen. May and June are transition months, so while you can expect mild temperatures to early Summer heat, extremes can happen.

I have been to Madrid twice, both times in May.

The first time we were on our way to the Canary Islands, and I was prepared for warm weather and beach time. Madrid,, situated on a high plain, was clear and sunny but cool enough I had to shop for a sweater for extra warmth. Now I have a nice black cardigan as a souvenir from Spain.

The second time was part of a longer trip around Spain. It was nice and sunny, but not hot, wherever we went, with no rain at all.

I know this is "anecdotal" but the lesson is to be prepared for anything, from cool to hot and rain and wind as well. Or be willing to go shopping.

May would be better for Madrid, but June better for the Dordogne. I've been to both in mid summer and while it was hot in Madrid it wasn't brutal, it will totally depend on your specific time there. Could be horrid, could be lovely. The Dordogne is the same in the sense that 'it depends'. But given that one place will generally be 'nicer' in May and the other in June I would probably start in late May. How long is your trip going to be.

Thanks everyone! I checked the sites for historical weather. It looks like May , particularly early May might be very nice for Madrid. So far, the Dordogne also looks warm enough for May. But I'll have to check a little further. I would imagine that pretty much all the markets are operating by May. I remember the markets in Beynac and Cenac were slowing down or stopping when I was last there in September. Don't think I've ever been to the Dordogne in spring so I'm unsure about markets.

We’ve always had good luck in Spain with weather and temperature in May. With the Madrid area there may be occasion at night and early morning where you might want a light jacket, but short sleeves during the day is mostly the norm. If windy, you may need that light jacket. By mid-June, for my wife and I at least, the temperatures are starting to get warm and it’s not as comfortable when doing a lot of walking around town.

I don't care for hot weather so I am willing to risk chill and rain traveling in Spring. When we were in Madrid in early May it was pleasant and not too hot -- but this time of year is always chancy.

I first visited Madrid and Seville in July and managed the heat pretty well. Of course, I am from the American Southeast and used to heat. My Daughter lives in Boston and I remember touring in Boston and Salem one June and temps were in the high 70s and I remember some of the New England people complaining about the heat. I suppose it is what you are used to living with that counts.