Has there been a case of a hybrid between a bottlenose and an Atlantic spotted dolphin?
Perhaps! No one knows for sure. There has been no definitive proof from DNA samples of wild animals, nor has it ever been confirmed in captive animals. However, there is much reason to believe it is plausible, and one possible hybrid has been observed.
Let me take you to The Bahamas. Specifically, Great Bahama Bank and Little Bahama Bank. They're sprawling sandbanks surrounding the islands, and many Atlantic spotted dolphins as well as coastal Common bottlenose dolphins make their home there. The spotted dolphins have been studied for more than 30 years by Denise Herzing and her associates. While by far the most encounters are with single-species groups, there have been dozens of mixed species encounters as well.
Many of these are aggressively sexual in nature, with males of both species sparring and trying to enter each other. Male bottlenose dolphins often seem to dominate the smaller spotted dolphins during these encounters, though the spotted dolphins have been seen retaliating, and it is speculated they don't actually experience the harassment as very negative. Since much of the action is male-on-male or involves juvenile animals not yet sexually mature, it seems in many cases the nature of these encounters is more social than sexual. However, multiple cases of interspecific male-on-female copulation have been observed.
With so much going on, the potential for hybridism is great. And since the two species are morphologically quite similar (especially when young), a hybrid could easily fly under the radar.
In February of 2003, Herzing and her team observed a female spotted dolphin with an unusual looking calf. The calf had a shorter rostrum than is normal for spotted dolphins in the area, but it was longer and more elegant than that of a Bahamian bottlenose. Herzing assumed it could be a hybrid calf. However, since no DNA testing was done and the calf was only observed for two days, this can't be confirmed. In 1989 and 1996 there have also been observations of female spotted dolphins mothering over bottlenose dolphin calves. In one case the calf was emaciated, and the spotted dolphin swimming with it wasn't lactating, so most likely the calf died. In another case it was only a temporary association and the bottlenose calf returned to their real mother after a while.
So a lot of interesting things going on. The studies are still ongoing. To my knowledge no large scale DNA testing has been done, nor any new "suspect" animals observed. Perhaps in the future a hybrid will be confirmed! Or perhaps it turns out this is one of those mixes that just doesn't work.
I highly recommend you read the study the photo comes from: "Interspecies interactions between Atlantic spotted dolphins, Stenella frontalis and bottlenose dolphins, Tursiops truncatus, on Great Bahama Bank, Bahamas". And many of the other studies done by Denise Herzing's team can be read for free here: https://www.wilddolphinproject.org/media/scientific-publications/