Yes, you can use both together. A web search of Project Gutenberg comes up with examples. Robert Louis Stevenson, in a letter, wrote "The place as regards scenery is grand, gloomy, and bleak." Washington Irving wrote "the river [wound] between tremendous walls of basaltic rock, that rose perpendicularly from the water’s edge, frowning in bleak and gloomy grandeur." A blogger writing about Charles Dickens' novel, "Bleak House," says "The story is bleak and gloomy. It’s set in foggy, dirty Victorian London, there is a lot of mention of dirt, squalor, disease, death and poverty." Precise meanings are hard to define and words are not always used precisely. "Gloomy" has the central meaning of "dark." It's fairly narrow and specific. It really refers not having much light. We can refer to the "gloom of night." It can also refer to a person's feelings or disposition: "he was a gloomy person," meaning someone with depressing, dark thoughts. "Bleak" is less well-defined, and regardless of word origin tends to refer to _bad weather,_ not only dark but also cold and windy. It often carries the idea of a place outdoors that is barren, without vegetation, exposed, and windswept. "Bleak" can also refer to the bad feelings you get from that kind of weather or that kind of place.

To understand the difference between these two words it is helpful to look at their etymological roots. "Gloom" originates with the idea of a darkness, twilight, or somber mood. The word continues to refer to negative mood. "Bleak", on the other hand, originates with brightness, whiteness, or blankness. From this original meaning, it acquired a meanings of bare, windswept, or cheerless. So, at least in reference to color, the words almost have opposite meaning. There is some intersection between the two words because a situation that is bare or cheerless can also produce in the beholder a negative mood. I would not use both words in one sentence though because they create somewhat opposite images. I recommend consulting an etymological dictionary to explore word origins. It is a useful tool.

'Bleak' is more extreme, and I would probably only use it in the winter (when it is cloudy/gloomy, there is a lack of sunshine, cold, and maybe windy). 'Bleak' implies that the weather is hostile and not friendly to life. 'Gloomy' is more general--I would use it to describe an overcast, cloudy day when there is minimal sunshine. It could also be sort of rainy or windy, but not necessarily so. 'Gloomy' is more just unfriendly to the mood.

In this context , they both mean the same. = cloudy and grey.

I would add that both words can refer to a person's outlook, as well. Gloomy mood or outlook - negative or pessimistic. (Her gloomy mood kept people away.) Bleak outlook - dire, without hope. (His future looked bleak.) Gloom is transitory, but a situation that is bleak is unlikely to change. Hope that helps.