Are spectra really the same as cohomology theories?

Consider the periodic complex $K$-theory spectrum $KU$. The integral homology group $H_i(KU)$, the direct limit of

$$\dots \to H_{2n+i}(BU)\to H_{2n+2+i}(BU)\to\dots,$$ is a one-dimensional rational vector space if $i$ is even and trivial if $i$ is odd. It follows that $H^1(KU)$ is nontrivial. (It's $Ext(\mathbb Q,\mathbb Z)$.) But this can't be detected in the cohomology of suspension spectra, because $H^{2n+1}(BU)$ is trivial.

So that's an example of a "hyperphantom" map from $KU$ to the Eilenberg-MacLane spectrum $\Sigma H\mathbb Z$.

The answer to this question is in LMS (I.6.9 of http://www.math.uchicago.edu/~may/BOOKS/equi.pdf) and in McClure's contribution to BMMS (VII\S1 of http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf), which gives full details. Let $T = \{T_i\}$ be a prespectrum. There is a cylinder construction $ZT$ that gives a weakly equivalent $\Omega$-spectrum (I first defined it in 1968 http://www.math.uchicago.edu/~may/PAPERS/7.pdf).
It is the telescope of the desuspensions $\Sigma^{-i} \Sigma^{\infty} T_i$. There is no loss of generality in taking $X=ZT$ in your question. There results a $lim^{1}$ exact sequence of the form

$$ 0 \to lim^{1}[\Sigma^{1-i} \Sigma^{\infty} T_i,Y] \to [X,Y] \to \lim[\Sigma^{-i} \Sigma^{\infty} T_i,Y]\to 0.$$

This has nothing to do with finite CW spectra, a priori, and it can be viewed via the usual adjunctions as giving a precise measure of the difference between the stable homotopy category of spectra and the homotopy category of based spaces. McClure (VII\S4 op cit) gives a clear criterion for when the $lim^{1}$ term vanishes and examples where the criterion holds. It is obviously not to be expected that the $lim^{1}$ term vanishes in general. One way to construct counter-examples is to relate this $lim^{1}$ exact sequence with the one given by approximating $X$ by a CW-spectrum, but I'll leave that to the interested reader. Of course, the elements of this $lim^{1}$ term are your hyperphantom maps.

Added as an edit: In answer to Tom Goodwillie's comment, the adjunctions I referred to give that if $Y$ is an $\Omega$-spectrum with $i$th space $Y_i$, then $$[\Sigma^{-i} \Sigma^{\infty} T_i,Y] \cong [T_i,Y_i]. $$ The brackets refer to spectra on left and based spaces on the right. Therefore the original $lim^{1}$ exact sequence can be rewritten as $$ 0 \to lim^{1}[T_i,Y_{i-1}] \to [X,Y] \to \lim[T_i,Y_i]\to 0.$$ The $lim$ and $lim^1$ terms are computed in terms of homotopy classes of maps of based spaces. That is what I had in mind with my sloppy statement about comparing homotopy categories. This is really a comparison between the stable homotopy category and the category of cohomology theories on based spaces, answering the original question.

Everyone go have fun: it's New Year's Eve (with a whole new meaning to the countdown to midnight).