Rainier first week of June, weather?

I was hoping to make a trip out to Rainier this year, but it looks as though the only time I could make it work is in the next 2-3 weeks. I understand that July/Aug is usually the best time to go, what are the drawbacks of going at this time of year? Snow storms and the cold? Should I put this off another year? I have equipment to stay warm well below zero, I'd rather not witness sub -10 degree weather as that is about the extent of the cold I have experienced.

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