The closest to bleeding out happened when I was about 7. We had one of those steel framed , vinyl liner wading pools. We were filling it when my younger brother took the hose and stuck it in a glass gallon milk bottle.
I grabbed it from him and as I tried to remove the hose, it exploded. Not only did it shred the pool liner, a glass shard cut my leg literally a 1/4" from the main artery. That took a handful of stitches to close it back up. the doctor told my Mom if it would have sliced that artery, I would have bled out in under a minute. You can still see the scar 54 years later.
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I was building a car model when the exacto knife I was using slipped off the model piece I was trimming and cut my left pointing finger to the bone. ( I would have cut it clean off but the bone was in the way.
)
You can still see the scar 50 years later.