Should you ever not assign an unique ID to each row in a table?
Yes, for example in case of a mapping table representing an N:M association between two tables. In its simplest form, it contains only two foreign keys, and these form its compound key, so it needs no separate ID column. In fact, adding an ID to such a table (without a uniqueness constraint, as pointed out by @Jeff) would allow one to add multiple associations between the same two rows, which is usually not desired.