In his student guide on page 154, Adams gives a construction of products for cohomology using "pairings" of spectra (now known as maps from $E\wedge E\to E$). But then he says

However, G. W. Whitehead did not say (because it is not true) that every product in generalised cohomology theory arises from a pairing in this sense.

On the other hand, if $E$ is a CW-spectrum (as in Adams' Chicago lecture notes), one can turn it into an $\omega$-spectrum in the category of CW-spectra using Milnor's theorem. Then each product in the corresponding generalized cohomology theory tautologically defines maps $E_k\wedge E_\ell\to E_{k+\ell}$.

Thus my questions: What examples have G. W. Whitehead and J. F. Adams been thinking of? In which situations would the naïve Argument above fail?