Can blood test with have different results depending on time of day they are taken?

Mom has Alzheimer's 82. No short term memory but is still functional but confused. Her thyroid T4 was fine a year ago but blood was drawn during the day when she had more energy. Had blood drawn again a year later early am no food and now her thyroid T4 is considered high 6.10 though T3 was fine. Can a blood test with thyroid differ in results as to when its taken? (AM or PM) or is a test a test? How can a thyroid go to Hypo within a year. Plus having problems getting her to take her pills she is not a pill person. When she did take them she seemed more energetic but in a way maybe too much so I wonder if she's really HYPO. Dr said her thyroid is getting old. Should Thyroid reading be the same no matter when its taken?