"У тебя была птица?"
Перевод:¿Has tenido un pájaro?
Right, but that happens with common nouns, where one can exclude the indefinite article when he/she wants it to be less concrete, and with a more general aspect. If you put it, then, the noun, the thing becomes more specific, it can specify the number ''or the noun itself''.
''¿tienes un pájaro? ¿O tienes dos?'' (here it specifies the number)
''¿tienes un lápiz? / ¿tienes lápiz?'' - Here even if both sentences mean the same, in the first one that article just makes the pencil to have more importance, since of all the items you can use to write you are specifying you want a pencil, not another thing.
The second one's intention is more vague, even if you are talking about a pencil, most of times it can be understood as wether you have something to write.
(Извините, но я не говорю по-русски).