My condolences to you on the loss of your mother. Unfortunately, proving that someone lacked the capacity to make a Will is pretty difficult, because the standard itself is actually fairly low. Just because your mother suffered a mental illness and may not have been following a prescribed medication regime does not mean she did not have the legal capacity to make a valid Will at the time she made hers, and if the attorney who prepared the Will for her was any good he or she should have taken a number of steps designed to help ensure that your mother was, in fact, competent to make the Will at that time and that it did reflect her actual wishes. If you believe that you have good, strong evidence that your mother really didn't have the capacity, or that someone from the charity was unduly influencing her, then you should consult an experienced estate litigator. That attorney should be able to help you estimate your chances of success with the challenge to the Will and the costs. You may find, however, that the projected expense of litigation, especially if the charity decides to fight rather than simply offering a settlement to make you go away, would quickly eat up enough to make your recovery, assuming you had one, a net loss. In addition, if your mother had a prior Will which also disinherited the children, a challenge to this Will could still result in your receiving nothing. But it does not hurt for you to talk to an attorney about it if you really believe you have a case.
I would also like to point out that your mother is completely free to disinherit her children (assuming she lived in Georgia at the time of her death and all of her children were adults, especially). You may be "heirs," but that means only that you would inherit if your mother had not done a Will, and that you have standing to challenge a Will if you believe it's not a valid expression of your mother's true intent. But that does not mean you have any right to receive assets from her if she took steps to make a Will and left her assets elsewhere. I'm not assuming you aren't aware of this, but just wanted to add it as a way to help clarify this point for others who may read your question and this answer. It's a common misconception that children have a right to receive property from their parents after the parents' deaths. In Georgia, this is not the case. The only thing even close is a year's support, and that's only available to children under 18, and has to be awarded by a court after a petition.
This answer is not intended to provide you with specific legal advice regarding your situation, or to create any attorney-client relationship. The intent is only to provide general information. You should be aware that you cannot rely on this answer to provide you with any protection against tax penalties. You should always consult your own attorney in order to obtain legal advice.
I agree with Ms. DiSalvo, who has given you a very good answer. I would simply add that, while you do have an uphill battle, it can be won, if you have compelling evidence. One of the bigger problems you face is that the lawyer will testify in support of the Will, and against your assertions. It may be that your mother had more capacity than you believe she did. As Ms. DiSalvo indicated, not a lot of capacity is required.
I second the suggestion that you consult with an attorney to review your facts and evidence.
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