**In rings**:  Let $R$ be a ring where idempotents lift modulo the Jacobson radical $J(R)$.  Any countable set of orthogonal idempotents in $R/J(R)$ lifts to an orthogonal set of idempotents; but this fails for uncountable sets.

**In combinatorics of words**:  Vaughan Pratt's "crossword problem" is another example.  Suppose you have a language consisting of words of length $\kappa$ in the letters $\{0,1\}$, satisfying the following three conditions:  (1) The all $0$'s word, and the all $1$'s word, belong to your language. (2) For any two distinct positions in $\kappa$, there is a word with $1$ in the first position and $0$ in the second.  (3) If you fill in a $\kappa\times\kappa$ crossword, so that every row and every column is a word in your language, then the main diagonal is also a word.

Pratt asked: Must your language contain all possible words?  The answer is yes if $\kappa\leq \aleph_0$, and no otherwise.