Yes, wind instruments can play out of tune, even when the instrument is "tuned properly" (which isn't as well-defined as it seems). In fact, the same can be said for fretted string instruments as well.
For wind instruments, the way you blow into the instrument can drastically affect your pitch. As a flute player, I can vary between as much as a whole step above and below the note I'm fingering by basically just rotating the instrument. I've heard single-reed (saxophone and clarinet) players slide up an octave without changing fingering. For brass instruments, you can get a lot of different pitches out of each fingering due to the nature of the instrument and the interaction with the mouthpiece. I would say, in general, that playing out of tune is the default situation for all wind instruments, and playing in tune requires a lot of practice and very good ear.
For fretted string, the amount of pressure you put on your fretting hand affects the pitch. It's not as much as an intentional bend, but it can be up to a quarter step or more, and definitely enough to sound out of tune even to untrained ears. This issue become much more pronounced with scalloped frets. It's not as much of a concern as wind instruments, however, because it's a universal solution: fret lightly and you will stay in tune, vs. wind instruments needing to take care of many different things for each note and situation they are in.
The instruments which cannot (without effort) play out of tune are non-fingered string instruments (harp, dulcimer, lyre), percussion instruments (drums, keyboards, auxiliary), and those instruments which are somehow both (piano, harpsichord, celeste). These instruments are all basically just a set of pretuned objects which are vibrated by picking, plucking, or striking to create the sound. Assuming those objects are in tune and you aren't going out of your way to touch or bend them inappropriately, they will play in tune. However, you can make them play out of tune in various ways, but it will not be an accident, and may be harder than playing in tune.