While a chest or large chest is opening, other block UIs (crafting tables, stonecutter, furnaces, brewing stands, anvils, other chests) can be opened. These will appear normally before being obscured by the chest interface, closing which reveals that they were still present, with a priority descending to the last opened.
Closing the anvil UI after closing the chest's will result in a crash. Tested in Survival Mode, with blocks placed in Creative mode.