No, it’s not.
You sent your Familiars to do a set of 4 Bounties.
The reason the Bounty Board is preventing you from acquiring another 4 Bounties is because otherwise you’d send Familiars out to do 4, then the hero would do two sets of 4, resulting in you getting 3 sets of 4 bounties.
–==[ EDIT ]==–
Took screenshots to demonstrate…
Here, I visit the Bounty Board, with 0/8 of my daily Bounties accepted…
After accepting the first four Bounties, the Board shows as 4/8 Bounties accepted…
I then visited the Familiar NPC and choose the Bounties option…
I send out the Familiars to collect the Bounties (6 hours)…
If I then re-visit the Bounty Board, it shows as 8/8 Bounties accepted…
That’s because my Hero has accepted 4/8 of them, and my Familiars have accepted 4/8 of them.