The Adams operations aren't always non-negative. (They are if you restrict to the subring generated by line bundles, though.)

Here's a counterexample in the world of finite CW-complexes. Let $X$ be the truncated projective space $\mathbb{RP}^{2n-1}/\mathbb{RP}^{2t-1}$. It's reduced $K$-theory was calculated by Adams [here][1] to be $\mathbb{Z}\oplus\mathbb{Z}/2^{n-t}\mathbb{Z}$. Let $\mu$ and $\nu$ be generators of each factor. They are best understood as follows. The inclusion of the $2t$-skeleton gives a map $S^{2t}\hookrightarrow X$. Pulling back along this map sends $\mu$ to a generator of $\tilde{K}(S^{2t})$. Let $\eta$ be the complexified tautological line bundle over $\mathbb{RP}^{2n-1}$. Pulling back along the quotient map $\mathbb{RP}^{2n-1}\rightarrow X$ sends $\nu$ to the class $([\eta]-1)^{t+1}=(-2)^{t}([\eta]-1)$. Note that the last equality is a simple consequence of the fact that real line bundles are self dual. Adams also calculated the effect of the Adams operations on these generators. For example, if $k$ is odd then $$\psi^{k}\mu=k^{t}\mu+\frac{k^{t}-1}{2}\nu$$
(Sidenote: that 2 in the denominator is the poison dart in Adams' vector-fields-on-spheres proof. For our purposes it is only important that $\psi^{k}\mu$ has a component in both the $\mu$ and $\nu$ factor. If we add a $t$ dimensional trivial bundle to the class $\mu$, then it is represented by an honest bundle $V$, which can be constructed via clutching, by cutting apart the $S^{2t}$ that comprises the $2t$-skeleton of $X$. Therefore,
$$\psi^{k}[V]=k^{t}[V]+ t -k^{t}t +\frac{k^{t}-1}{2}[\eta]  -\frac{k^{t}-1}{2}$$ 
 Multiplying the clutching function of $V$ by $k^{t}$ (as an element in $\pi_{2t-1}U(t)=\mathbb{Z}$ gives a vector bundle $V_{k^{t}}$ representing $k^{t}\mu+t$, so
$$\psi^{k}[V]=[V_{k^{t}}]+\frac{k^{t}-1}{2}[\eta] -\frac{k^{t}-1}{2}$$ 
Now, it's a general fact that if $E$ is a vector bundle such that the class $[E]-n$ is representable by a vector bundle, then the last $n$ Chern classes of $E$ must be zero. We can achieve this obstruction in our example by choosing the right $k$, $n$, and $t$. The first term has a single non-vanishing (top) chern class, which is $k^{t}$ times a generator of $H^{2t}(X)=\mathbb{Z}$, and the chern class of the second term is easily understood in terms of the chern class of $\eta$. In particular its top chern class will be nonzero as long as $n$ is very large.

  [1]: https://apurvanakade.github.io/notes/CMI/vector%20fields%20on%20spheres/adams2.pdf