Fetch data between a period in a table with start date and end date fields

I'm having a problem fetching data in a table with start date and end date fields between a period (a filter from-to)

The sample data is:

ID - INICIO     - FIM
1  - 01/01/2017 - 31/12/2017
2  - 10/01/2017 - 01/02/2017
3  - 01/05/2017 - 01/06/2017
4  - 15/05/2017 - 15/06/2017
5  - 01/09/2017 - 31/10/2017

Dai I have 3 searches and with expected returns

1st search between 01/01/2017-31/12/2017
expected return-1-2-3-4-5

2nd search between 01/04/2017-30/06/2017
expected return-1-3-4

3rd search between 02/06/2017-02/07/2017
expected return - 1-4

The query I made for first search was: http://sqlfiddle.com/#! 9 / ba8d19/2

select * from dados
where
    dt_inicio >= '2017-01-01' AND dt_inicio <= '2017-12-31'
AND dt_fim >= '2017-01-01' AND dt_fim <= '2017-12-31'
// o resultado foi ok

The query I made for second search was: http://sqlfiddle.com/#! 9 / ba8d19/3

select * from dados
where
    dt_inicio >= '2017-04-01' AND dt_inicio <= '2017-06-30'
AND dt_fim >= '2017-04-01' AND dt_fim <= '2017-06-30'
// o resultado FALHOU

The query I made for third search was: http://sqlfiddle.com/#! 9 / ba8d19/4

select * from dados
where
    dt_inicio >= '2017-06-02' AND dt_inicio <= '2017-07-02'
AND dt_fim >= '2017-06-02' AND dt_fim <= '2017-07-02'
// o resultado FALHOU

Follows script to create table and data:

CREATE TABLE IF NOT EXISTS `dados` (
`id` int(11) NOT NULL,
  `dt_inicio` date NOT NULL DEFAULT '0000-00-00',
  `dt_fim` date NOT NULL DEFAULT '0000-00-00'
) ENGINE=InnoDB AUTO_INCREMENT=6 DEFAULT CHARSET=utf8;


INSERT INTO `dados` (`id`, `dt_inicio`, `dt_fim`) VALUES
(1, '2017-01-01', '2017-12-31'),
(2, '2017-01-10', '2017-02-01'),
(3, '2017-05-01', '2017-06-01'),
(4, '2017-05-15', '2017-06-15'),
(5, '2017-01-09', '2017-10-31')