The projections $\mathcal{P}(\mathcal{C}(H))$ in the Calkin algebra have a similar property with the pair "on the other side", i.e. for every countable subset one can find a pair with exactly the same upper (or lower) bounds.  This can be seen from the proof of Theorem 4.1 of my paper [Filters in C*-algebras][1].  Conversely, for any pair (or even countable subset) one can find an increasing sequence with exactly the same upper bounds, which applies more generally to the projections in any C*-algebra with real rank zero.  As with Turing degrees, this can also be used to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice.  To see this first note that $\mathcal{P}(\mathcal{C}(H))$ has no $(\omega,1)$-gaps, by essentially the same argument used with $\mathscr{P}(\omega)/\mathrm{fin}$.  Thus to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice one simply needs to find a pair of projections that has a strictly increasing sequence with the same upper bounds, as done by Farah/Hadwin/Weaver (see [this mathoverflow post][2])


  [1]: http://arxiv.org/abs/1109.6077
  [2]: https://mathoverflow.net/questions/172073/why-do-the-projections-in-the-calkin-algebra-not-form-a-lattice