This is extremely common, in fact, you are more likely to find dungeons connected to mineshafts than caves, especially larger ones; for example, these are all the dungeons within an area, with over half of them connected to a couple mineshafts (upper-right and lower-left, these are two separate mineshafts):
Note that this is a vanilla 1.6.4 world but they have always generated like this, well, unless Mojang ever fixes this "bug" (if they do then dungeons will become much rarer, or would if mineshafts hadn't already effectively become 5 times rarer since 1.6.4, including 2.5 times less common overall since 1.7 and with double the ground to generate in since 1.18, and only the parts within solid ground may have dungeons):
(the only form of this that might warrant being fixed is dungeons attached to or inside of strongholds, desert temples, etc; mineshafts are much more natural given the lack of proper walls, and in fact, I simply view them as a variant of cave)
SEED : 0
Coordinates : -169 -29 -94
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This is extremely common, in fact, you are more likely to find dungeons connected to mineshafts than caves, especially larger ones; for example, these are all the dungeons within an area, with over half of them connected to a couple mineshafts (upper-right and lower-left, these are two separate mineshafts):
Note that this is a vanilla 1.6.4 world but they have always generated like this, well, unless Mojang ever fixes this "bug" (if they do then dungeons will become much rarer, or would if mineshafts hadn't already effectively become 5 times rarer since 1.6.4, including 2.5 times less common overall since 1.7 and with double the ground to generate in since 1.18, and only the parts within solid ground may have dungeons):
MC-186343 Dungeon intersects mineshaft
(the only form of this that might warrant being fixed is dungeons attached to or inside of strongholds, desert temples, etc; mineshafts are much more natural given the lack of proper walls, and in fact, I simply view them as a variant of cave)
TheMasterCaver's First World - possibly the most caved-out world in Minecraft history - includes world download.
TheMasterCaver's World - my own version of Minecraft largely based on my views of how the game should have evolved since 1.6.4.
Why do I still play in 1.6.4?