Given a non-commutative non-type-I factor $M$, if $P(M)$ denotes the lattice of projections in $M$ (with the usual ordering $x \leq y \iff$ range$(x) \subseteq $ range$(y)$), then every nonzero $x \in P(M)$ has an uncountable antichain of projections below it. (@martin_argerami gave a nice proof of this here.)
Since any antichain below $x$ can be extended to a maximal antichain below $x$ -- also called a "partition" of $x$ -- there is an uncountable partition of $x$.
Question: are there cases where some projection $x$ also has a nontrivial countable partition in $P(M)$? (Again, assuming $M$ is a noncommutative non-type-I factor; and nontrivial meaning not the partition consisting just of $x$)