Is it only possible to pay attention to one thing at a time?
The teacher is correct. 

It is only possible for the mind to hold a single object at a time. Thinking that the mind is able to hold multiple objects at the same time is merely an illusion of continuity. It is just the mind shifting very rapidly between objects and mindfulness is not yet strong enough to register/discern that.

As mindfulness, concentration and insight matures that fact will gradually become more apparent.

By practicing Vipassana meditation one can come to clearly experience this fact.