What do you associate with such a heavy snow?
What do you associate such a heavy snow with?

Not really sure what you are trying to say above.  From Cambridge dictionary:

associate (CONNECT)
to connect someone or something in your mind with someone or something else:
Most people associate this brand with good quality.

associate with sb phrasal verb
to spend time with a group of people, especially people who are disapproved of:
I don't want my children associating with drug-addicts and alcoholics.

If I had to pick one, I'd go with the second one.  But then an expected answer could be "orange sherbert or Italian ice or ice cream," or something in that order.

Perhaps you are trying to ask why did such a heavy snow fall?   Then I would rephrase the question as "To what do you attribute such a heavy snow fall?" Then an expected answer could be "accumulation of moisture in the atmosphere colliding with a cold front...."

From Cambridge:
attribute sth to sb/sth (RESULT) phrasal verb
to say or think that something is the result or work of something or someone else:
The doctors have attributed the cause of the illness to an unknown virus.
To what do you attribute this delay?
Most experts have attributed the drawing to Michelangelo.