From experience with my own kids as well as asking some other moms and doing some research on popular baby websites, it seems to be the average time a baby sleeps in their bassinet is approximately 3 months or around 9 weeks – 13 weeks. It is important to check your bassinet for specific guidelines and weight limits.There are some bassinets that have a weight limit of 10 pounds while others may be as high as 20 pounds. Another thing to consider besides the weight limit is how long your baby is as well.My first son was quite skinny but very long.By the time he was about 6 weeks old he was almost as long as his bassinet.nother thing to consider besides the weight limit is how long your baby is as well. My first son was quite skinny but very long. By the time he was about 6 weeks old he was almost as long as his bassinet.