Hi Seth,
No, the replacements were all one after another within a very short time period, all of it under the original warranty.
The warranty is not machine specific, it is time specific; as part of the purchase agreement, they are committed to "keep you whole" one way or another for a period of 12 months, no matter how they decide to do it. Logically, the same printer can be repaired and break down again in month 13 and you would be S.O.L. on a 12 month warranty, so it's really no different than if the replacement printer were to break down in month 7 of its life with you starting 6 months from the time you sent back the first machine. So again, I wouldn't lose sleep over this possibility - it's the same risk one way or another. If they were to send you a refurb that they failed to refurb properly, you would know that well in time to get it replaced.
Now as to the problem itself, if they've advised you in a way suggesting that it's a hardware problem, I would take their word for it. Perhaps try upgrading or replacing the firmware to see whether that helps. If it doesn't, and the problem persists, I would think it best to follow their advice.