If the website states that there is a 5 day turnaround for orders, then being patient means that you wait for an extra couple of days before getting bothered.
I'm currently coming up on day 22 (or 15 business days) for the shirts that I ordered back in October. I'm bothered.
Managing expectations is pretty critical in business, so my recommendation to Luxire at the moment would be to double or triple their posted delivery times for their shirts - that way people are pleasantly surprised when the shirts arrive early, rather then annoyed that they have to send emails every week to find out why their order is late.
That is true, but common sense dictates that, no matter what the website says, there will be occasions through the year where the advertised times are not achievable. Think about it, you're getting MTM shirts, at a very high standard, for not a great deal of money. Waiting a little bit longer, on occasions, is hardly going to hurt you, nor is it reason for complaint here. Exercise some common sense and patience, at times, and everything will be fine.