Yes -- all the people I know withdrawing from small banks are going to Chase or Morgan Stanley. These largest banks have to mark-to-market their losses and have a lot of cash (short duration) assets, and are literally too big to fail.
Confusingly, the two banks you named are JP Morgan Chase, and Morgan Stanley. JP Morgan Chase is the biggest bank there is, but Morgan Stanley is far smaller (but still ginormious) and doesn't rank in the top 10 banks. The top 3 banks are JP Morgan Chase, Bank of America, and Citigroup.
Within JPM Chase, the "JP Morgan" or "JP Morgan Private Bank" brand is used for (V)HNWI banking and the "Chase" brand is for retail (although Chase does have a Chase Private Client arm for almost-HNWIs). Due to the nature of catering to VHNWIs, "JP Morgan" of course offers wealth management services.
Morgan Stanley is a completely different company with no (current) ties to JPMC.
The reason JPMC and M-S share the name "Morgan" is because the investment and retail sides of the bank were split during the Great Depression due to Glass-Steagall, almost 100 years ago.
And, more importantly, are national banks regulated by the occ so they can't do the same kinds of risky things that svb & co can do. (They can do other risky things that they come up with historically, especially pre dodd-frank, but not this)
