What is the difference between $D(\mathbb Z)$ and $Spectra$ in terms of $t$-structures? I'm trying to see my way around the following

False Claim: Bounded spectra are the same as bounded chain complexes (as a triangulated category, say).

Dubious Proof: Consider the standard $t$-structure on $Spectra$. The bounded objects in this $t$-structure are those spectra with finitely many nonzero homotopy groups, which forms a full triangulated subcategory with an induced $t$-structure, which is bounded. The heart of this $t$-structure is the category $\mathbb Z \text{-}Mod$ of abelian groups. The functor from bounded $t$-structures to abelian categories sending a $t$-structure to its heart is an equivalence. Therefore we have an equivalence between bounded spectra and the bounded derived category of abelian groups.
Questions:


*

*Where did I go wrong?

*Is there some triangulated-category-theoretic concept which "measures" the difference between chain complexes and spectra?
 A: As Qiaochu points out, I'm simply wildly wrong in believing that a bounded $t$-structure is determined by its heart -- although if you fix an ambient triangulated category $\mathcal T$ it is true that the bounded part of any $t$-structure on $\mathcal T$ is determined by its heart.
As alluded to in the comments, there is a recognition principle for determining when a $t$-structure is the derived category of its heart, given in Lurie's Higher Algebra, Prop 1.3.3.7 (used to spectacular effect by Gheorghe, Wang, and Xu). There is always a functor from the derived category of the heart to the original triangulated category. Lurie gives a recognition principle for when this functor is fully faithful and identifies its essential image. In the case of spectra, it boils down to asking whether $Ext_{Spectra}^\ast(\mathbb Z, \mathbb Z)$ vanishes for $\ast > 0$. Which it doesn't, because there exist nontrivial stable integral cohomology operations. Stable integral cohomology operations are not that familiar, so I'm a bit more comfortable $p$-completing and observing that the forgetful functor from $H\mathbb F_p$-modules to $p$-complete spectra is not fully faithful because the Steenrod algebra at $p$ is nontrivial.
