Yes, T cells learn in your thymus what is self and what is not. The way they learn is both positive and negative (isn't that a contridiction) and by the strength of their ability to bind - they have to know what is self to know what is not self during an infection. If a T cell can't recognize self weakly, then it dies. If it recognizes self too strongly, then it dies. Thus, your T cells have to know self, but not too well.
During an infection, T cells recognize foreign antigens (or initiators of the immune response) in context of self. What that means is that the foreign antigen is digested by special phagocytic cells and then combines with the proteins that T cells recognize as self. The T cell will bind very strongly to the foreign antigen/self complex. It then initiates a cascade of other events that will kill a virus infected cell as an example.
When an organ is transplanted, the antigens are the same, but the self molecules are not. Thus rejection - same response, but for the opposite reason.
Aplasitic anemia is caused by some drugs such as the antibiotic chloramphenicol, but as you haven't indicated that, then you make have what would be more likely something called idiopathic. When doctors call something idiopathic it means they know know the actual triggering cause! If I can speculate some about your case, let's turn to MS and see how it compares. We know that multiple sclerosis is caused by T cells turning against self as a result of a virus or bacterial infection. The T cells react to self because of what we call a cross reactive response to self - the infection triggered a response to a foreign antigen that "looks" a lot like self. How foreign is foreign? Sometimes it is only one amino acid in a whole antigen protein and if much of the rest of the protein is the same as one we already have, then cross reactivity can result. MS, type I diabetes and many other diseases are triggered by some infection leading to a cross reactive response against self antigens. Certainly, we know there are genetic components. I would speculate that you may have had such a triggering even and we would likely never truly know what it is.
There are a few reasons why we may never know...viruses in particular, and some bacteria that cause infections in us have evolved the ability to look like us - if you can mimic the host, you are less likely to get elimiated by the host's immune response. Others have well known ability to change their coat, making it very difficult for us scientists to figure out vaccines - malaria and the common cold for example. When we can't figure out the antiges that the infectious agent makes, we have an even harder time then determining what triggered your specific response. Then to top it off, your self molecules are very different from anyone else's (the genetic component), thus you present antigens differently from anyone else, so we rarely can use a genetic test to predict who may have this type of problem.
I understand the severity of your affliction and applaud your approach to understanding it. I trust you are on a list for a bone marrow transplant, perhaps through the unrelated bone marrow donor program. I hope this bit of explaination also shows you some of the issues finding matches outside your family. The self proteins are very diverse to protect humanity's survival. We need such diversity to be able to mount an immune response to just about anything! Thus there are hundreds and hundreds of different possiblities for 6 major genes, hence the complexity of find unrelated bone marrow donors that match.