I'm making a slightly stronger point than may immediately be apparent, which is that for some of these strange Banach-space properties (a famous example being the property of not containing $c_0$ or any $\ell_p$ space, which was first shown to be possible by Tsirelson), not only is there considerable flexibility in how you build counterexamples, but it appears that this flexibility is in some sense "necessary". One way of making that assertion semi-precise is to say that there don't seem to be additional (sensible) properties you can insist on that cause the flexibility to go away.