No.
The Fed is raising rates. They raised rates yesterday. They say they intend to keep doing so through the end of the year. Net interest is a low single digit percent of the federal budget; it's lower as a fraction of GDP than it was in the 90s [1]. Most of the federal debt is fixed rate--raising rates now only affects future borrowing.
The real limit on rates is growth and employment. If the economy falters because people are spending all their money on servicing debts over goods and services, we'll see a crunch. That's not happening. The opposite is happening: inflation is surging.