No, the logistics of that would be baffling.
Students cannot possibly belong to more than one house. They are placed in both their classes and assigned to a dormitory by house. In the house cup, they also earn points for or get points deducted from their house for great success or failure.
“Welcome to Hogwarts,’ said Professor McGonagall. ‘The start-of-term banquet will begin shortly, but before you take your seats in the Great Hall, you will be sorted into your houses. The Sorting is a very important ceremony because, while you are here, your house will be something like your family within Hogwarts. You will have classes with the rest of your house, sleep in your house dormitory and spend free time in your house common room.”
- Harry Potter and the Philosopher's Stone, Chapter 7 (The Sorting Hat)
If it were possible for students to belong to multiple houses, then Hogwarts would have to find a way to determine which house they will share each class with, how often they sleep in each of their houses’ dormitories, and which of their houses should be added to or taken from in points if they earn or lose points in the House Cup. Therefore, it would be far more logical to only allow students to be sorted into and then belong to one house.
Students can have traits of other houses.
However, students may still have strong traits of other houses than their own. Hermione had sufficiently strong Ravenclaw traits that the Sorting Hat considered having her be one.
“How come you’re not in Ravenclaw?’ he demanded, staring at Hermione with something close to wonder. ‘With brains like yours?’
‘Well, the Sorting Hat did seriously consider putting me in Ravenclaw during my Sorting,’ said Hermione brightly, ‘but it decided on Gryffindor in the end.”
- Harry Potter and the Order of the Phoenix, Chapter 11 (The Sorting Hat’s New Song)
However, he decided on Gryffindor, so Hermione is a Gryffindor. Though she still has strong Ravenclaw traits, from the time she was sorted, her house became Gryffindor and Gryffindor alone. Her Ravenclaw traits would still be present as strongly but they would have no effect on her Hogwarts house.