This is not a simple question. It will depend on several factors including relevant state and federal regulations as well as your private agreement with them.
A lawyer would need to review any and all relevant info in order to make that determination. That said, if you only stood to gain a modest return for that period this may not be worth that effort and expense.
How have they proposed to handle it? Did they acknowledge they made a mistake or did they have an alternative explanation?
You may want to discuss with a lawyer in private. Most of us here, including myself, offer a free phone consult.
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As a FINRA arbitrator, I can tell you these are not simple matters. You need a lawyer experienced in this type of work who will want to see every document you signed and every email. Your proof of damages will be difficult unless you specifically advised them of what to invest in. Your contributory negligence in waiting for 3 months to check your account may come into play. Finally, licensing issues of an online investment firm in the state where you reside may come into play.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
The answer is it depends. The first question that I would need to know is is you dealing with a broker-dealer or investment adviser. The answer to that question will determine what sort of duty they owe you. Also, what sort of diligence did you show in reviewing activity in the account? Do you have any record of the transaction that was supposed to occur? Often, these disputes are decided by who has the best records..
My colleagues have all nailed this down but I have one question you need to ask yourself in terms of the evidence you have of their wrongdoing. You state "During this time the investments that should have been made on my behalf increased in value" .... My question is do you have written proof that you ordered a purchase of specific securities to be made that was not executed? if you do not have this tied down to the tee then you are going to have a difficult time for sure.
My answer is not intended to be giving legal advice and this topic can be a complex area where the advice of a licensed attorney in your State should be obtained.