I don't know how this same misinformation gets resurrected every year, but SRNA's at BCM do NOT, EVER share cases with a resident. In the beginning part of each academic cycle, when most trainees are new and rotating at the county hospital, SRNAs may be paired in rooms together, and new CA1s (especially during 'mentor mode') may be paired together or with an upper level resident. But never do the two mix. The only times SRNAs and residents may mingle on the same patient is with a hand-off at the end of the day when the call team arrives, or on-call if the call team includes SRNA(s). In the latter instance, SRNAs are assigned cases throughout the night while supervised by the senior resident, or may lend an extra hand for stat intubations by the senior or stat cases (e.g. trauma) where any available person helps.
The most important thing to be 'wary' about is taking advice from internet forums. If interested in these programs, investigate them directly with people who know (chief resident emails can be accessed from most programs' internet sites, and they wouldn't mind answering simple questions like these). Much better than 2nd or 3rd-hand information.