I have health and skills to get to day 5, but I now exceed the "Carry limit" and my dweller is turned back. Does anyone get to day 6 and beyond in the wasteland?

Just raise every SPECIAL stat possible, give them a real good weapon and hope for the best I guess. Apparently power armour helps too, but that might just be because of its boost. I think it all comes down to stats, level and weaponry

EDIT:

I think if you build multiple fully upgraded stimpak rooms, your dweller will be able to carry a LOT more stimpaks and therefor last longer in the wastelands. I'm not too sure though

I think they mean that the dweller fills up with items, not that they die, stimpaks are capped at 25 that you can give to someone going into the wasteland regardless of how many you can hold in the vault(think there was an exploit that let you get more way back that was patched, but regardless) I think luck affects how many items you get, (or maybe it's just the quality? don't know for sure) so a high str/end character with low luck may be able to stay out there a fair while. still, I'm pretty sure there's a point where the quality of items that you get stops going up, though I don't know when.

Before the October update, I had explorers doing 6 days and more. Got auto recalled right after the update.

The max item an explorer can collect is 100. Even the exploration continues, no more items can be collected. So it is good to have them recalled rather than continue.

Some players suggested Bethesda give the option to drop the normal, that would Help to get better gear.

No, you cannot make it to day five or six anymore. 100 items usually comes between 4 days 5 hours, and 12 hours. Once they hit 100 items they turn around. Wish we could tell them to drop the common items like we can sell all the common when they get back.

There are new http://www.gamerevolution.com/faq/fallout-shelter/pets-full-list-of-dogs-and-cats-in-fallout-shelter-120985 that lets you carry 6 to 29 more items(depending on the animal), and that would give you enough additional hours to get well past 5 days(probably not 6 though).

I had a legendaries child (40 stat) level 1 trained to stat 70, sent out with confessor rag and 21 weapon, 25 stimpaks. He passed 6 days 6 hrs but got recalled due to the October update.

If you have similar dwellers and correct pet, you can try it.