IRS says I owe $3k interest but is issuing me a refund?

I received a notice from the IRS that I miscalculated my tax due for 2014 and am owed an $8k refund, but that they are only sending me $5k because they say I owe $3k in interest from a balance due from 2011.

I haven't received any notification that I ever owed anything. I've processed a tax return each year - and have held over the over-payment to next year's return each time. I've never owed anything to my knowledge.

I've ordered a transcript and will go to my local office to find out what's going on.

In the meantime, my question is… If I owe a tax balance large enough to have $3k of interest… How are they sending me a $5k refund? Wouldn't they just keep the refund?

Is it normal for people to have tax balances that they don't know about? No bill, no notice, no past-due notice, no wage garnishment, no lien on my house, no problems getting a mortgage?
Added (1). Thanks! I've been at the same address 20+ years and have a secured mailbox. The amount of interest they say I owe is the exact amount (to the penny) that I calculated my 2014 over-payment to be on my 1040. Something just seems off.

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