I think given the right circumstances it would be possible to sneak into Aman before it was detached from the world. In fact, EarendilEärendil himself did it.
We get our first glimpse of the difficulty of getting to Aman when EarendilEärendil tries to find the Valar and ask them to come to the aid of both Eldar and Edain in middleMiddle-earth.
Eärendil found not Tuor nor Idril, nor came he ever on that journey to the shores of Valinor, defeated by shadows and enchantment, driven by repelling winds, until in longing for Elwing he turned homeward towards the coast of Beleriand.
- "Of the Voyage of Eärendil and the War of Wrath," Chapter 24 Of the Voyage of Eärendil and the War of Wrath/Chapter 24 TheThe Silmarillion
So here we get our first glimpse of the answer. EarendilEärendil could not find Valinor even with Vingilot the ship built with the aid of Círdan the greatest ship builder in middleMiddle-earth, and an elf.
Of course EarendilEärendil was eventually able to find Valinor but only when he posessedpossessed one of the Silmaril.
..but sought back once more to Valinor with Elwing at his side. He stood now most often at the prow of Vingilot, and the Silmaril was bound upon his brow; and ever its light grew greater as they drew into the West. And the wise have said that it was by reason of the power of that holy jewel that they came in time to waters that no vessels save those of the Teleri had known;
- "Of the Voyage of Eärendil and the War of Wrath," Chapter 24 Of the Voyage of Eärendil and the War of Wrath/Chapter 24 TheThe Silmarillion
The reaction of the Valar and EonweEönwë the herald of ManweManwë reveals to us that this was entirely unexpected:
So I think we can see here that the coming of EarendilEärendil was unexpected and he had indeed "snuck" into Aman. This is a very specific case though as the only reason he made it to amanAman was his possession of one of the Silmarils.
For a more concrete answer though we look at the tale Ælfwine the angloAnglo-saxon who found the straight road and sailed to Aman from Wales. A rather interesting story which can be found in chapter 6 of the Book Of Lost TalesBook Of Lost Tales: part 2.
Ælfwine stumbled upon the straight road and ended up in Aman, with no magic to prohibit him going though. Of course, these does hit a snag as Ælfwine can be translated to ('elf-friend') and as some other users have pointed out, may be a kind of 'free-pass' and Ælfwine was a distant relation of EarendilEärendil.
(I need sources for this but don't have my book with me, so see tolkiengatewaytolkiengateway )
now there is the possibility that this was allowed by Manwë or IlluvatarIlúvatar for the following reason: