The philosopher who is most famous for advocating both the bundle theory of objects, and phenomenalism, is the 18th century Irish philosopher, George Berkeley. Berkeley's version is more commonly called "subjective idealism".
Philosophers who hear the skeptic's challenge -- "There's no reason to think an external world exists" -- reply, "Well, no, I guess there isn't any reason to think that an external world exists. All there is, is sense-data. Physical objects are bundles of sense-data. When I hold up my hand, and I see it, I'm not seeing something external to my mind; I'm seeing a series, a whole bundle, of hand sense-data, and there is no hand apart from those hand sense-data. That's what my hand is -- a bundle of sense-data." Such philosophers get around skepticism, not by replying to the skeptic and proving the existence of an external world, but instead by saying that there is no external world.
One objection to phenomenalism, uses a reductio ad absurdum. Suppose the phenomenalist is correct. This is the assumption we intend to disprove.
Phenomenalism means that, for me, the physical world is all just a construction out of my sense-data. Now suppose you and I are talking about philosophy. On the face of it, there's my body, and my mind associated with it, and there's your body, with your body associated with it. But we are assuming that phenomenalism is true. That means that when I see your body, I'm not seeing an irreducibly physical body; I am seeing a bundle of sense-data in my own mind. Let's suppose I hear you saying all sorts of intelligent things, which I want to take as evidence of the existence of your mind, of what you are thinking. All those intelligent things you say are, after all, sense-data in my own mind. So I have no reason to think that either your body, or your mind exist. The phenomenalist has no reason to believe that any other minds, besides his own, exist.
Why should the phenomenalist be surprised when we say that? After all, phenomenalism denies that an external world exists. Just remember what "external world" means according to the phenomenalist: it means the world outside his own mind. But that means that all other bodies and other minds are part of the external world. And so the phenomenalist is forced to an absurd conclusion. The argument in brief is:
Phenomenalism would force us to accept solipsism, so we had better reject our assumption that the phenomenalist is correct.
-Edit- Despite the inclination to reject solipsism out of hand, its argument is made in such a way that it is that it is not falsifiable and therefore cannot be disproven; at least not by means of modern science, philosophy included. By extension phenomenalism is not disproven by the above argument in any way.