Hi, Gretchen. I was in Scotland the last two weeks in June and I never wore a t-shirt, though my husband did. It wouldn't hurt to bring a couple in case you get a heat wave, but you will probably need long sleeves. Plan on layers. I wore a long sleeve t-shirt, fleece and waterproof windbreaker almost every day. I would like to echo the comment that a waterproof jacket is a must, preferably with a hood. Northface makes great jackets that are waterproof and have hoods that keep your head completely dry and do not blow off in the wind. Also, the hoods do not obscure your vision. They are not cheap, but they are ideal for a climate like Scotland's. We had rain almost every day, but most days it was just a very light rain. I would also suggest pants that dry fairly quickly -- anything but jeans.
p.s. We were in Scotland in May once and it was a bit colder. I wore a cotton turtleneck as my base layer on that trip. I don't think you will need that if you have a fleece, though.