This is true if you are planning to reapply for the same card, because the automated system may decline the application if it sees that you already hold the same personal card. Closed cards stay visible for about a month, but downgraded cards for only a few days.

The decision to move or close the credit line associated with a card is a matter of strategy. Are you comfortable calling Chase reconsideration, asking them to move credit to create a credit line for a new card, and answering their other questions?