I write fantasy, and am considering writing one of my worlds as having a few stars that are faintly visible during most of the day–obviously more or less so depending on time of day, season etc. But I know very little about astronomy, so basically, would it be possible for a planet with an atmosphere similar to Earth’s to be close enough to other stars for them to be visible in the daytime, or would that make some kind of gravity snafu? Thanks!
Venus is often visible in the daytime, if you know where to look; so any star brighter than Venus (about mag- 4) should be visible in the daytime. This could easily happen - if we were ten light years or less away from one of the brighter stars, for instance Deneb or Rigel, we could see it in daytime with relative ease.
Another way to see stars in the daytime would be to be near, or in an open cluster or globular cluster; the stars in these locations are very close together and this should make for a fairly spectacular sky.
Even better would be the sky in the Galactic Hub, although such a very crowded location would be a dangerous place for an Earth-like planet. Crowded locations tend to have more supernovas, and if one occurs near your planet it can cause a mass extinction or worse.
You can see Venus during the day. So, as long as your stars have an apparent magnitude of roughly -4 you’d be fine.
So say your star is roughly 4 light years away, it would then have to have a luminosity roughly 50x of the sun.
OP seems to be focused on actual stars, such as other stars in their galaxy – not planets in their solar system.
Pretty sure the answer is yes, if the planet in question is in a solar system which has some very bright nearby neighbor stars in the right phase of life.
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Just to reinforce Grey’s post, many of the familiar stars in the sky are more than 50x as bright as the Sun. Deneb is one if the brightest, and is about 160,000 times as bright as Sol; Rigel is about 120,000 times as bright, and there are many others. So any planet within a few tens of light years of these beacon stars would see them easily in daylight.
Yet another way to look at this is by considering the absolute magnitude of some very bright stars. The absolute magnitude is the brightness of the star at a distance of 10 parsecs, a distance which corresponds to 32.6 light years.
On this list
every star with an absolute magnitude greater than -4 would be visible during the day; that is every star above (and including) Canopus in that list. (remember that -5 is brighter than -4, and so on).
This is by no means an exhaustive list, by the way.
Perhaps the most likely scenario for a star visible during the day would be a binary system, where the binary star is very far away from the central star (so as not to perturb the Earth-like planets). Even with modest brightness, it would still be quite close astronomically speaking, but also far enough away to be a point of light that would look like any other star. There are known binaries with a star out in Pluto’s orbit or farther.
I think it’s the magnitude of Venus that matters, not the type of object. It is demonstrably true that an object with a magnitude of -4 is visible from Earth during the day. Therefore a star with the same magnitude would also be visible.
Next step is if a star would have to be close enough to earth to have a meaningful gravitational impact before it would be close enough to have that magnitude.
Or if not a gravitational impact, some other sort of direct effects that would preclude life. Stars that bright would be high mass, short lived, and they come to a violent end. Dense clusters of stars going supernova like firecrackers every hundred million years or so would tend to put a damper on the evolution of complex life.
You could probably even get away with something fainter than Venus. Stars are point sources, while Venus is a visible disk. Having the light all concentrated in a point would make it easier to see.
In practice, you’d probably want something a fair bit brighter than Venus, because while Venus can be seen in the daytime, it’s really hard, not something you’d notice if you didn’t look right where to look. With bright enough stars, though, or stars close enough together, it’s still not implausible.
That’s really pushing it.
For all practical purposes, to the naked eye Venus IS a point source.
Yeah yeah yeah there might be some 10 percent effect or something something but that is about it.
PS. Are you ever going to revisit your coffee shop tutoring thread ? 
It’s different for the practical purpose of twinkling, at least, which is the only practical purpose I can think of here.
Would a very subtle change in the atmosphere of Earth make the sky significantly less bright, and hence many more stars visible in daytime? What characteristic of the earth’s atmosphere renders the sky so bright in daytime, and is there any change that would make the sky darker, while stilll leaving the atmosphere “similar to Earth”. Our atmosphere is constantly changing (ozone layer, for example, or particulate). Have stars ever been visible from the Earth, prior to relatively modern times, owing to slight changes in the atmosphere, man-made or otherwise? Could dinosaurs see stars in daytime, before the atmospheric event that led to their extinction? Are there species of animals with eyesight that enables them to see stars in daytime today?
I’ve seen Jupiter in a bright twilight blue sky, and Venus in broad daylight.
The sky could be full of stars in daylight in several locations in our galaxy.
According to this, 1984JBAA...94..221H Page 221, Sirius is visible during the day on real-life earth, albeit only under particular circumstances and knowing exactly where to look.
Some people claim they can spot Sirius right before dusk, when it has been first located for them. Takes 20-10 vision, apparently.
We live in a rather out of the way corner of the Milky Way, thinly scattered with stars, and under even the clearest of skies cannot see any stars in the daytime, with the possible exception of Sirius.
But there are definitely areas in the galaxy, closer to the center, where this definitely wouldn’t be true. From the awesome and free Celestia planetarium program, just for illustration purposes, I downloaded this list of brightest known stars as seen from the so-called Pistol Star, which is about 25K LY from us, in the direction of Sagittarius.
I’m only showing the first page, but scrolling down I found about 40 stars of magnitude -5 or brighter, which would presumably be visible in the daytime skies of a planet orbiting a star in this region. I say “in this region”, because such a planet most likely couldn’t be found orbiting the Pistol Star or anything like it. This is a hypergiant–an extremely bright, unstable, and short-lived star that should only last a few million years, so life would not have a chance to evolve on any planet orbiting it.
But if we assume it’s somehow possible for a solar type star to exist in this region, together with a planet at the right distance to support life, then these extremely bright stars in the list would certainly be visible by daylight–most likely even in that planet’s equivalent of L.A.
The problem with living near the galactic centre, in close proximity to so many bright stars, is that bright stars explode at the end of their short life. There would be so many supernovae near a planet orbiting a sun-like star in that region that you’d have a nearby supernova every couple of million years - this would cause frequent mass extinctions, and make the course of evolution on such a world extremely different to that on Earth. Stars in the galactic centre are generally considered to be outside the so-called Galactic Habitable Zone.
Of course, any planet “in the region” of the Pistol Star would also have the Pistol Star itself in its sky.
The sky is bright (and blue) mainly because sunlight is scattered by oxygen and nitrogen molecules (Rayleigh scattering). It can only be reduced by making sunlight dimmer, or the air thinner. But either change would probably make the planet uninhabitable. (If the sun were much dimmer, the planet would be much colder.)