No, you do not have a case against the prospective employer:1) One way to view what transpired is that you rejected the employment offer of starting more-or-less immediately and countered to the prospective employer with an offer to start in three weeks; counteroffers reject the original offer. In that case, there was no offer from the prospective employer on the table.2) Even if it were viewed that you had already accepted the initial offer of employment, the fact that you asked to delay your start date would provide  grounds to terminate your employment (e.g. withdraw the offer), since you went to the prospective employer and said that you did not want to begin work immediately; indicating to an employer that you would have problems working the agreed-upon schedule  is grounds for termination.3) Morever, an offer of employment generally does not form an actual employment contract unless the offer restricts the ability to terminate in some way--for example, it guarantees employment for a year. If this offer did not form an employment contract, you would have been an employee at will and could be terminated (offer withdrawn) at any time, for any reason.