That is nonsense.  Someone with no idea of how things work just tossed a number.

Once the average traffic (transactions issued per day by all users) exceeds the effective capacity of the network (about 240'000 tx/day, 2.8 tx/s), there will be a steadily growing backlog of unconfirmed transactions.  There will not be a fixed delay: the average delay will keep increasing, day after day.

If the traffic were to be 300'000 tx/day (3.5 tx/s), for example, the backlog would grow at the rate of ~60'000 tx per day (~0.7 tx/s).  The transactions received in one day will take 300'000/240'000*24 = 30 hours to be confirmed; so the average delay for 1-confirmation will keep growing by 6 hours every day.  

If the transactions were serviced first-in, first-out, then all transactions sent in the same hour would be delayed by about the same (but always growing) number of hours.   Since the processing priority is largely determined by the fees paid, however, some transactions may be processed in the next block, independently of the backlog, while others will be delayed even more than they would with the fair policy.   However, there will be no way to estimate the fee needed to get your transaction confirmed in the next block, or within X hours; because that depends on the fees of transactions that will be issued before your confirmation -- by clients who will want their transactions to be processed before yours.

But that regime cannot last for long, of course.  Clients will give up on bitcoin, until traffic drops below the capacity -- say to 220'000 tx/day (~2.6 tx/s). Then, part of the time, there will be no backlog: all transactions will confirm in the next block, even if they pay the minimal fee.  However, the traffic varies a lot depending on time of day and day of the week.  During peak hours and peak days, the traffic will be quite a bit more than 2.8 tx/s. Then there will be a temporary backlog, lasting several hours, that will be cleared slowly when the traffic subsides.  

As before, the backlog and the average confirmation delay will keep growing while the traffic exceeds the capacity.  The delay for your transaction, specifically, will depend on its fee, on the transactions that are waiting in the queue, and on the transactions that will be issued by other clients until your transaction is confirmed.  On some occasions, many transactions may get delayed by several hours, perhaps by a day or two.

Let me put this another way:

I bought bitcoin because it never even occurred to me that it wouldn't scale.  Many new users don't know now that it doesn't scale, but when we run up against the hard limit and your beloved transaction fee market kicks off, everyone will know and nobody likes paying high fees for something they are used to getting nearly free.

The investors/speculators are the only reason this thing works at all.  You are taking us for granted.  We do not exist simply to facilitate drug deals.  WE buy the coins. WE supply the liquidity. If you don't give us what we want, we invest elsewhere.  Without us, nobody pays the miners and everything grinds to a halt.