Attorneys and clients sometimes don't agree and can't get on the same page. When that happens it's time for a parting of the ways. Unless a client discharges an attorney for cause, the attorney is entitled to be paid for his or her work up to the time of the discharge. Discharge for cause would be for misconduct such as a conflict of interest or gross inattention to the case. You have not described any such problem. When the fee agreement was contingent, the outgoing attorney generally gets a share of the fee at the end of the case. You did not provide enough information for me to guess whether 1/3 is reasonable in this instance. You need to hire another lawyer. When you interview other lawyers, I'm sure you will discuss what they think the outgoing attorney should get.
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By your reference to "1/3" I am assuming that this is a personal injury case. If so, a lawyer who terminates representation may or may not have a statutory lien against your future settlement/judgment proceeds, depending on the nature of the termination. usually, it is the client who terminates the representation. the statutory lien is not defined in terms of a percentage, even if the fee agreement with the attorney was a contingent percentage fee. rather, it is defined as a reasonable fee for the services performed to date, plus any out-of-pocket expenditures made on the case by the lawyer. In every case I have handled where there was a previous lawyer asserting a lien, or in the two cases I can recall ever asserting a lien, the lien amount was agreed to among old and new counsel, so as not to affect the client's share of the proceeds (although that is not what is legally required--the law provides that the discharged lawyer has a lien against the client's share; however, my practice and that of every lawyer I have dealt with is as I have described). Disagreements or communication issues among lawyer and client arise from time to time, so your situation is not unique. I would have your new lawyer negotiate with the old lawyer regarding her lien. If she refuses to budge on her 1/3 demand, insist that she provide an hourly breakdown of the dates and time spent and services provided on your case. A reasonable hourly fee can be charged. If the time or services are unreasonable, then you can refer the matter to DC Bar Counsel, or seek mediation.
You've left out one crucial piece of information here, namely whether anything has been filed by this attorney with the court. If this attorney has already made the representation to the court that she represents you, the only way that she can withdraw is if a judge allows her to. There are only two recognized reasons for withdrawing from a case: irreconcilable differences and extreme financial hardship. If this attorney is the attorney of record for you case but has not asked the court whether she may withdraw, you are in a better position as to whether you would owe her anything. If she is your attorney of record and has told you that she is withdrawing without permission from a judge to do so, then she has effectively committed the ethics violation of client abandonment.
I am very suspicious of your statement ". . . my attorney decides to abandon my case because of her lack of comprehension to the conversation. . . ." It sounds to me that you two have a bad, bordering on hostile, relationship. That fact does not mean that proper steps can be avoided to end the attorney/client relationship.
My advice should in no way be construed as creating an attorney-client relationship.
Your attorneys right to represent you must spring from a contract, usually a written one. The terms of the attorneys representation should be governed by the written document/contract. Attorneys should not take on cases that they are not competent to handle - Rule 1.1 of the Rules of Professional Conduct. An attorney who wishes to withdraw from further representation may do so under most circumstances (see 13th amendment to US Constitution) but there are often rules imposed by the court that advise that the attorney is required to give you formal written notice and an opportunity to retain new counsel so that your case is not adversely impacted by the withdrawal.
As for the "old attorney" the presumption is that you had a written agreement - which governs the terms of fees. If it's a contingency fee contract, then the attorneys' right to collect 1/3rd should be based upon there having been a recovery. If there was no recovery, then there is likely nothing owed by you. That said, attorneys may have a right to collect money for the value of services provided - quantum meruit. If all the original attorney did was refer you to someone else and that someone else did little or nothing and then withdrew as your counsel, it seems as if there was very little "value" provided to you. An attorney who gets fired by his client can try to sue the client for the value of the services provided. However, suing a client is probably not good for the reputation of the attorney and if there was no meaningful value provided to the client by the attorney then the damages (the claim value of the services) that might be sought would be so small as to make a lawsuit undesirable or not feasible.
Rule 1.4 of the Rules of Professional Conduct require the attorney to carefully listen to the client so as to be able to comply with reasonable requests for information and so as to be able to explain the matter to the extent reasonably necessary to permit the client to make informed decisions about the case or about the representation itself. Basically, a client can't give informed consent until the lawyer has communicated adequate information and explanation about the material risks of and reasonably available alternatives to the proposed course of conduct.