Police say the attacks in the capital and Puli Khumri were targeting government employees.
Bomb attacks in Kabul and northern Afghanistan have killed at least four people and wounded 13 others.
Wednesday's blasts came as the country marks its 101st Independence Day, the celebrations for which were held a day earlier.
Two sticky bombs targeted government employees in the Afghan capital, killing two people, including a police officer, and wounding two others, police said.
In Puli Khumri, the capital of northern Baghlan province, a bomb targeted a vehicle belonging to the provincial intelligence department, killing two service members and wounding 11 people, including both military and civilians, Nazir Najem, the provincial governor's spokesman, said.
No one immediately claimed responsibility for the attacks.
Kabul police spokesman Ferdaws Faramarz said one of the sticky bombs was attached to a police vehicle, while the other was attached to a car belonging to the education ministry. The police were investigating, he said.
Taliban spokesman Zabiullah Mujahid said he was not aware of the explosions in Kabul.
Daesh group claimed responsibility for a mortar attack the previous day in Kabul that killed three people, including two government employees, and wounded at least 16, including four children and woman.
Tariq Arian, the Interior Ministry's spokesperson, said militants fired a barrage of rockets from two vehicles toward different areas of the city.
Earlier, reports said only that there were 10 wounded.
Daesh said it fired 16 mortar shells, targeting the presidential palace, embassies and Afghan government offices in Kabul.
Authorities said most of the shells hit residential homes.
The violence comes amid new uncertainties over the start of talks between the Taliban and the Kabul political leadership. The government said it would not release the last 320 Taliban prisoners it holds until the insurgents free more captured Afghan soldiers.
The decision went against that of a traditional Afghan council held earlier this month — the Loya Jirga — and is likely to further delay intra-Afghan peace talks sought by the United States.