Just my opinion here Richard but I've been to Yellowstone many times towards the third week of May and the weather was fine. The crowds are less which is a big plus.
There is still snow up at Canyon Jct and the higher elevations but so what?
Sure the weather can be anything that time of year but I don't know if I'd let just that one issue scare you away.
The bison migrate from the lower valleys to the upper valleys as spring turns to summer. Often if you arrive in mid june they might be in the middle of the migration and not yet in places like the hayden valley and you wont see any.
The weather is definitely warmer in July but the crowds are three times as thick too so just keep that in mind when deciding what is best for you.
Another great time of year is the fall after the kiddies go back to school. Say the first 2 weeks of September. The crowds thin out but its still warm (ish) (warm can be a relative term when talking about Yellowstone)