Yes and no. You can’t decline the increase and continue to use the credit card. However, if the rate increase is applicable only to future purchases—in other words, it is not a penalty increase, the go-to rate after an introductory rate expires, or the new variable rate based on a change in the index—you could opt out and close the account. This means you would not be able to use the card and the card issuer could increase your monthly payment to speed up the repayment process at the existing rate (either requiring you to pay off your balance in five years, or doubling the percentage of your balance used to calculate your minimum payment). Another option would be to simply stop using the card, without notifying the issuer, and continue to pay off the balance under the regular payment schedule.
You can’t opt out of a penalty rate increase, or an increase due to either the expiration of your introductory rate or a change in the index used to determine your variable interest rate. You could stop using the card so that you would not accrue new debt at the new, higher rate. But you would still be charged that higher rate as you pay off the existing balance.