He spoke so quickly that I can't understand him at all.
He spoke so quickly that I couldn't understand him at all.
What is more correct?
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"Can't" is present tense. It means you are having trouble understanding him right now.
"Couldn't" is past tense. It means you had trouble understanding him at some time in the past.
The first sentence is not correct as you are using the past tense "spoke" with the present tense "can't". The present tense version would be:
He is speaking so quickly that I can't understand him at all.