First, since $ \lfloor x + y \rfloor, $ $ \lfloor x \rfloor, $ $ \lfloor y \rfloor $ are all integers, \[\lfloor x + y \rfloor - \lfloor x \rfloor - \lfloor y \rfloor\]must also be an integer. We can write \[x = \lfloor x \rfloor + \{x\},\]where $ \{x\} $ represents the fractional part of $ x $. Similarly, we can also write $ y = \lfloor y \rfloor + \{y\} $ and $ x + y = \lfloor x + y \rfloor + \{x + y\}, $ so \begin{align*} \lfloor x + y \rfloor - \lfloor x \rfloor - \lfloor y \rfloor &= (x + y - \{x + y\}) - (x - \{x\}) - (y - \{y\}) \\ &= \{x\} + \{y\} - \{x + y\}.\end{align*}Note that $ 0 \le \{x\}, $ $ \{y\}, $ $ \{x + y\} < 1, $ so \[\{x\} + \{y\} - \{x + y\} > 0 + 0 - 1 = -1\]and \[\{x\} + \{y\} - \{x + y\} < 1 + 1 - 0 = 2.\]Since $ \lfloor x + y \rfloor - \lfloor x \rfloor - \lfloor y \rfloor = \{x\} + \{y\} - \{x + y\} $ is an integer, the only possible values are 0 and 1. For $ x = y = 0, $ \[\lfloor x + y \rfloor - \lfloor x \rfloor - \lfloor y \rfloor = 0 - 0 - 0 = 0,\]and for $ x = y = \frac{1}{2}, $ \[\lfloor x + y \rfloor - \lfloor x \rfloor - \lfloor y \rfloor = 1 - 0 - 0 = 1.\]Therefore, the possible values of $ \lfloor x + y \rfloor - \lfloor x \rfloor - \lfloor y \rfloor $ are $ \boxed{0,1} $.