Cat naps aren't good.. It's a sign of exhaustion.. Their little brains needs frequent rest to build synaptic connections and process and store information... I'm not a fan of letting them cry it out either... My parenting style is attachment parenting... Babies do not self-regulate... They depend on their caretakers to help them regulate their needs (hunger, sleep, diaper changes, baths, etc).... My daughter, who is 10 months old, has a schedule based on her biological needs.. She naps like clockwork at 10am-11am and again at 1-3:30, then goes to bed around 8 and sleeps till 8. She is fed on demand. The only food that is scheduled is solids. I feed her solids with the family at 8:30am, noon, and 6pm. She gets snacks between meals like my toddler.
How many ounces of formula/milk is she getting in a day? A baby her age should be getting between 26-32oz a day. If she's not getting enough food that could be an issue too. Babies will hit growth spurts and suddenly require extra food. My brother's wife, who is fat-phobic, would only give her infant the minimum ounces recommended per day. The baby needed more and would be fitful and fussy around nap time, napped poorly, and would wake up screaming (from hunger). Didn't matter what I said- his wife refused to change things and said the doctor said she would be fine and kept doing what she was doing.......
The 4 month old should be taking several hour long naps during the day- usually 3 naps, and she *should* be sleeping 12 hours through the night if she weighs more than 12lbs.
Check out the book "Healthy Sleep Habits, Happy Child", and "Happiest Baby on the Block"