Can the sum of two mixed numbers be equal to two

Not unless one of the mixed numbers was negative.  In order to make 2 (without using 1 + 1) one of the numbers would have to be greater than 1, and one would have to be less than 1. The one that was less than 1 wouldn't have an integer and therefor wouldn't be classified as a mixed number. However, if one of the mixed numbers was a negative value  < -1, both mixed numbers would have an integer and it would be possible. So if negative values are included, the answer is yes, but if negative values are excluded, the answer is no.