Is a $\Omega$-Spectrum a connective one? I can't find this result anywhere, but it seems pretty straightforward. I want to avoid silly mistakes, but I can't see any fault. I'd love to receive some feedback
Let $X$ be a $\Omega$-spectrum (of compactly generated topological spaces). By definition for $q<0$, $$\pi_q(X)=\lim_k \pi_{k+q <k}X_k$$
But since $\pi_i(X_k)\cong \pi_{i}(\Omega X_{k+1})\cong \pi_{i+1}(X_{k+1})$ the spaces are more and connected, in particular, $X_k$ is $k-1$-connected and therefore $\pi_q(X)=0$.
Is that correct?

A $\Omega$-spectrum is not necessarily a connective one.  In your argument, there is no reason "the spaces are more and [more] connected".  For a concrete example, consider the $K$-theory spectrum $KU$.  In its usual incarnation (with $BU \times \mathbb{Z}$ even and $\Omega(BU \times \mathbb{Z})$ odd), it is already a $\Omega$-spectrum by Bott periodicity.  But it's 2-periodic, so its homotopy groups extend infinitely in negative degrees.  
In general, remember then every spectrum can be converted into a $\Omega$-spectrum by a telescope construction, without changing its homotopy groups.  So if every $\Omega$-spectrum was connective, this would mean then every spectrum is connective, which should strike you as patently false.  This gives another bunch of counterexamples: take any spectrum, desuspend it a bunch of times so that it has negative homotopy groups, and then "$\Omega$-spectrify".
