For months the teen-ager and her baby lived a nomadic existence, moving among relatives and friends. But the 14-year-old cared for her son feeding him properly and taking him for medical checkups.

A social worker at the hospital where the baby was born began wondering why the baby had not been brought in for follow-up care, and a search, involving the city, was begun, for the teen-ager and her son.

Last April the city found the girl and here infant living with a friend of the girl. The baby was taken from the mother, and the girl's mother ordered her daughter to return home - without the baby.

Now the 1-year-old boy is in a foster home and his mother is living with her mother, waiting a judge's decision on the placement of her son.

While no two such situations are ever identical, the girl's situation illustrates the problems faced by many teen-aged mothers who for want to keep their babies: There are homes for the infants and there are homes in Washington for unwed mothers. But there are no homes, either institutional or foster, for mother and child together.

The teen-aged girl's mother, a 39-year-old government clerk, refused to discuss her family problems, but her daughter said:

"She feels that it's my responsibility and I agree with her on this much. She shouldn't have to take care of him. She went through enough with four of us and she don't want to be tied down with no baby. She feels if I bring my son in there she's going to be tied down and I'm going to be running the streets. But that's not right. That's not right!"

The D.C. government, which took the teen-aged girl into court on charges of child neglect, maintains that she has no home for the baby and therefore cannot care for him.

It also is the city's position that a parent has the legal right to tell a minor child where to live and the girl's mother thus has the right to order her daughter to live at home - without her baby.

"The baby was in great physical shape" said a city official familiar with the case. "I can't possibly say she took great care of the baby but I can't say anything to the contrary. But let's face it - a 14-year-old child on her own can't really make it. She needs help to take care of her child, right?

"We're not talking about a case where there's a bad guy. DHR (Human Resources Department), hasn't failed. The probation office hasn't failed. It's a Catch-22 situation. The grandmother won't take the baby into the home, but the grandmother isn't a villain completely either, although it's easy to make her one. She's a lady who has raised her own family and doesn't feel she's up to the responsibility of an infant. Nothing says we have to raise our grandchildren; only our children."

Nancy Polikoff, attorney for the girl, argued that "a juvenile has the same constitutional right to bear and raise children that an adult has. Her mother is putting her into a Catch-22 situation and there's nothing she can do to get out of it."

According to the girl, she and her mother "was getting along fairly well" prior to the girl's becoming pregnant by her 23-year-old boy friend. "We wasn't talking much," she said of her relationship with her mother with whom she lived, "but when we saw each other we would talk . . While I was pregnant we talked once and a while. After I had my son we didn't talk at all.

"She asked me was I going to have an abortion. I told her 'no.' She asked me what kind of means of support did I have. I told her his father was going to support him. She told me that the baby couldn't stay there."

The girl said the father offered to marry her, "but I didn't want that. I didn't want to marry him because it wouldn't have worked out."

Why did she turn down the abortion her mother suggested?

"She didn't have no abortion with me, so I feel like I shouldn't have no abortion with my son . . . I'm against it completely. I just don't like that sort of thing. I don't dig it.

"Since I did it, I might as well suffer the consequences," said the girl, who is now 15. "Just going and getting rid of it, that's kind of hard to forget something like that. Anything like that you'll remember for the rest of your life.Then you look back . . . and you say you wish you kept it.

"I ain't got no means of support, but I'm willing to try on my own to take care of him. I did it for 3 1/2 months. When they got my son he was very healthy. He showed no signs of sickness or nothing. If he was to stay (in a foster home) until I was 18. I'll see him as much as I can and take him to as many things as I can.

"But I can't do but so much for finances," said the Shaw Junior High School eighth-grader, "being the age I am, except for in the summer time because I can get a summer job.

"The problem is that my mother does not want my son in the house. She wants me there. She don't want my son. And I want to be with my son.

"I don't understand it really, because that's her first grandchild and mostly anybody would take care of their first grandchild," said the girl. "I don't see why she would do a thing like that. Turn her grandchild away. I don't see why I have to stay there when she didn't want my son there."

D.C. Superior Court Judge Gladys Kessler is expected to rule on the case in about two weeks. In the meantime, both the city and the young mother are faced with the problem that while there are facilities for teen-age mothers, and facilities for the children of teen-aged mothers, there are no institutions or foster homes for mother and child.

"I make no bones about it," said Nancy Polikoff, "I hope that through the publicity somebody will come forward and say I want a teen-age mother and her son . . . I refuse to believe there's nobody in the city who doesn't want to do this."