Yes, and this is part of what’s called the interest rate risk of bonds.
If someone purchased a $1,000 bond with a 5% coupon, and a year later, the company issued new $1,000 bonds with a 4% coupon, in order to buy the 5% coupon bond from the owner, you would obviously need to pay more than $1,000 (since the new bonds issued by the company have a 4% coupon).
What does Nominal Value mean?
What Happens to the Price of a Bond After I Buy It?
Can You Sell a Bond for Less Than the Price You Paid For It?