Silicon Valley Bank was the 16th largest bank in US.
Signature Bank was the 29th largest bank in US.
Collapse of a bank has direct and indirect impact on other companies and banks. Signature bank collapsed partially because of SVB collapse. Too many depositors withdrew their money at once.
Within a few days, the 14th largest bank in US, First Republic Bank is impacted by the collapse of Silicon Valley Bank. Large US banks provided $30 billion to stabilize First Republic Bank.
So, yes, the collapse of Silicon Valley Bank or another bank, can affect other banks. But the impact depends on how big is the bank and what is the structure of its assets.
In general, bank collapse may result into a financial crisis. The collapse of SVB and Signature Bank will probably not cause the world financial crisis, similar to the one in 2008, because these banks were not so large. But it may cause some type of local or regional financial crisis. To avoid or minimize potential financial crisis, US banks should take action to stabilize the banking system.