Yes. There can be a bloodshot appearance to the eye, meaning that the blood vessels on the white part of the eye are dilated and fuller looking than normal. Infections, inflammatory conditions, acute glaucoma, allergies, dry eye, corneal abrasions and trauma / foreign body are some of the more common causes of this kind of red eye.
Another form of red eye can be where the white of the eye looks uniformly red like a red spot or hemorrhage. This can be in a focal area, or throughout the white of the eye. This type of red eye comes from a hemorrhage from a small blood vessel on the surface of the eye.
In actuality, red eye is not a specific diagnosis, but a physical finding which indicates pathology in the eye.