The headline of the question is already the question itself: Is a spectrum a colimit of shifted suspension spectra?
By a spectrum I mean a sequence of spaces $E_n$ indexed over the natural numbers and structure maps $\Sigma E_n\to E_{n+1}$. A map of spectra are maps in all degrees commuting with the structure maps.
Yes. Any spectrum should be the colimit of its finite subspectra (which can always be taken in the form you require).