If we solely consider speeches about Hamlet by other characters, it would be easy to come to the conclusion that he is, in fact, mad. Polonius, in particular, describes him this way:
Madam, I swear I use no art at all.
That he is mad, 'tis true: 'tis true 'tis pity;
And pity 'tis 'tis true—a foolish figure!
Ophelia seems also to view him as mad, perhaps because her father is warning her of precisely this. Polonius blames Hamlet's apparent madness on his unrequited love for Ophelia, and much of the central portion of the play, especially the second act, is dominated by Polonius's efforts to determine what it is that is making Hamlet act so strangely. If we only read what Polonius, Gertrude, and Claudius have to say about Hamlet, we would get the rather shallow view that he is fully mad, and we would not begin to scratch the surface as to the true reasons for his behavior.