The reason Herodotus believed it was false is the same reason some modern historians believe it might be true: according to the account, the sailors saw the sun to their right as they sailed clockwise around southern Africa. This implies that they crossed the equator, which in Herodotus’ time was believed to be so hot as to be physically impassable.

(We do have another, more reliable account of a Carthaginian expedition that explored the coast of west Africa from the other direction, but apparently turned back just before crossing the equator.)

Note: for most of what follows I'm drawing on Richard Miles' Carthage Must be Destroyed

Short answer: probably not. However there are some interesting stories passed down to us about Carthaginian expeditions in Africa (Carthage began as a phoenician colony). One anonymous greek text from the 5th century bc called the Voyage of Hanno describes a sizeable fleet of 65 ships which set off south down the west coast of africa. Supposedly making is as far as the Niger delta and Mount Cameroon before they turned back. There is considerable debate as to whether the voyages actually happened. There is archaeological evidence that the phoenicians we're trading as far west as the coast of modern day Mauritania, but beyond that we're just left with this one account (which is suppose to have been copied from an inscription in Carthaginian temple). If there had only been one expedition that far south, we might expect a lack of archaeological evidence, still the open Atlantic Ocean is much more difficult to navigate than the (comparatively) peaceful Mediterranean - it proved difficult for Europeans even thousands of years later.