No; this is not the law.
The Seventh Circuit's decision in Brunker v. Schwan's Home Service, Inc., 483 F3d 1004 (7th Cir. 2009) shows why.
The plaintiff, Brunker, was a route manager for Schwan's, a home-delivery food service. Brunker began experiencing medical problems, which led to a diagnosis of multiple sclerosis, and was fired for alleged "unsatisfactory performance" and being "unable to perform essential job functions." He brought claims under the Americans With Disabilities Act (ADA) for disability discrimination and failure to accommodate.
Brunker sought production of the personnel files of other route managers who had been accused of performance deficiencies similar or like his own. Schwan's opposed these efforts and stated that it "would rely only on Brunker's failure to service customers as the reason it fired him," a position rendering, it claimed, irrelevant any other discipline that Brunker and/or his co-workers received. A magistrate judge accepted the employer's position and with prejudice, imposing $4,423 in sanctions on Brunker for seeking the discovery. The Seventh Circuit reversed not only the denial of discovery and the sanctions order explaining:
An employer accused of discrimination cannot limit discovery simply by restricting during litigation its stated reasons for an adverse action. After all, the true reason behind the action is the very heart of the case, and Brunker presented evidence that Schwan's may have asserted reasons for firing him other than the one it relied on during litigation. An employer's shifting explanations are evidence that its stated reasons did not truly motivate the adverse action and that an impermissible one actually did.
The Court's analysis brings to mind the Sixth Circuit's in its case, Bobo v. UPS, discussed at Similarly-Situated In All Relevant Respects; The Sixth Circuit Reverses Based on the District Court's Limiting Plaintiff's Discovery.