Yes, but you'd call it "malpractice," not "negligence." You should first start with a complaint to the state bar association - they regulate attorneys.
This answer is not to be considered a response to a specific legal issue in a specific jurisdiction - it is to be considered only a general response to a hypothetical scenario posed by the questioner. For specific legal advice, please consult with a licensed attorney in your jurisdiction.
You can sue for malpractice if you can afford to. These are technical difficult cases requiring you to prove that but for the attorney's acts and omissions a different result would have been obtained. This is an exceptionally high standard of proof. The difficulty of these cases is reflected in the very few attorneys willing to take them on contingency. So, if you have years to spend and thousand of dollars to commit to the effort, yes, you can sue. The suggestion to make your complaint to the state regulatory authorities (State Bar) instead may be very sound.
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