There are acoustical reasons for not wanting close voicing in the lower register; in short, the upper harmonics muddy each other up and fog up the sound.

But in my experience, C4 is a really high limit; I can think of *tons* of scores with thirds below C4.

Every musical environment is different, and sometimes you might want that slightly muddy sound. But if I had to come up with a hard rule, I would be inclined to use C3 as your upper limit instead of C4.

Below C3, composers often just stick with octaves, but occasionally you'll find some perfect fifths between tuba and bass trombone, or split between two tuba parts, etc.

Edit: As I mention in a comment to mkingsbu's answer, Mahler 3 has consecutive thirds of D2–F2–A2 at m. 209 of the first movement. These tight spacings do happen, but typically they're written to create a very specific effect.