Yes; it can. If one were to have a slow exchange of blood going throughout his/her body, then it is entirely possible for that individual to feel mentally distant, often lacking perception and acuteness. However, I'm not so sure as to whether or not low blood pressure can truly be blamed for states of tiredness as much as it can for an inability to concentrate. If the subject were to move quickly from one position to an other (i.e. Laying down to standing up), then the lack of blood (oxygen) to the brain can go so far as to cause fainting.
