The phrase like a dog with a bone means:
– tenacious, persistent, obstinate;
– unwilling to yield, to relent or to let go;
– unable to set aside a preoccupation or obsession.

This phrase refers to the fact that a dog with a bone will not let go of that bone, no matter what.

The phrase like a dog with a bone occurs, for example, in an interview of the British actor and singer Adam Rickitt (born 1978), who had played the part of Nick Tilsley in the British television soap opera Coronation Street—interview by Mark Taylor, published in the Evening Post (Bristol, England) of Thursday 10^th June 1999:

“When I joined Granada, I always said I didn’t want to be in The Street for a long time. I didn’t want to stay and become the next Ken Barlow, but it was still quite scary to leave.”
Although he may be away from acting for a while, Adam doesn’t rule out a return to the profession, as he still feels he has a lot to learn.
“My mum always says I’m like a dog with a bone because I don’t give up. I know I wasn’t that great at acting and I want to improve.”

The earliest occurrences of the phrase like a dog with a bone that I have found are as follows, in chronological order:

1-: From Splendid Follies (London: Printed for J. F. Hughes, 1810), a novel by an unknown author:

The second day of the sport, old Milford, finding himself indisposed to attend the race, like a dog with a bone, would not permit any of the family to enjoy that species of amusement, of which the inability of age forbid him to partake, and the visitors, in compliment, remained at home.

2-: From an article on “the question of the Presidential election”, published in The United States Gazette (Philadelphia, Pennsylvania, USA) of Tuesday 3^rd February 1824:

We now notice extracts from letters, notices of meetings, &c. once more in the public papers, and calculations are again renewed, by all parties. […]
[…]
Mr. CLINTON—But he is not considered a candidate, yet the very possibility that his friends might advance his claims, has brought down upon him the ire of some politicians, that appear very much like a dog with a bone—who while he holds fast his filched prize, snarles [sic] and growls at every passenger that he thinks may interfere with his ill-acquired possession.

3-: From The Anti-Reformers.—A new Song. To the old Tune of “Which Nobody can deny.”, published in The Morning Chronicle (London, England) of Monday 7^th March 1831:

There’s Cr—r, who snarls “like a dog with a bone”—
(If the simile vexes—’tis one of his own)—
So snappish, so surly, so vulgar in tone,
                    Which nobody can deny.
If some people fancy to give him a kick,
You could not much hurt him—his skin is so thick:
It serv’d him in stead when he wrote THE PIC-NIC,
                    Which nobody can deny.
He, too, laments deeply the people’s hard case,
And puts on for the mob a most pitiful face;
But out-of-place pity is quite out of place,
                    Which nobody can deny.

4-: From a song published in The Literary Gazette; and Journal of Belles Lettres, Arts, Sciences, &c. (London, England) of Saturday 17^th August 1833—song reprinted from Soap-Bubbles; or, the Lyrics of Robert Olden (Cork (Ireland): 1833):

For good-humoured faces, Cork once beat all places,
How altered the case is, more a thrue mavrone!*
By politics now are contracted each brow, or
Every nose turned up sour,† like a dog with a bone.
* We believe an Irish expression of grief: Mr. Olden has given us no explanation.—Ed. L. G.
† Pronounced in Ireland sow-er.—Ed. L. G.

5-: From the Nantucket Inquirer (Nantucket, Massachusetts, USA) of Monday 23^rd June 1834:

[For the Inquirer.]
THE FAT MAN.

          “Oh that this too, too solid flesh would melt,
           Thaw and dissolve itself into a dew!”
There are some men on this earth who are born fat—who live and die fat. Let them grumble till the “crack of doom,” or the rip of their breeches, yet it availeth not; and they might as well think to drain out Niagara with a clamshell, as bring themselves down to the common, every-day proportions of humanity. Nature laughs at their strainings, and pulls one way while they pull another. They roll though life, puffing and wheezing, and putting the best face they possibly can upon their rotundity. Now and then, however, when a little vinegar gets mingled with the milk of human kindness, your true fat man is the most sensitive, morose and unmanageable biped that draws one foot after the other. He snaps like a dog with a bone, at every body.

6-: From the Columbian Register (New Haven, Connecticut, USA) of Saturday 25^th November 1837:

Vice-President Johnson.—To show that federalism never forgives the sin of democracy, we need only allude to the insulting course adopted by the whig press in New-York, during his late visit to that city. We have never seen so unprovoked and abusive an article in the New-York Journal of Commerce, as the following. The federalists are so flushed with their late accidental victory, produced by causes which will evaporate before the next election, that they cannot exercise the common decencies of life towards an opponent—not even towards one who has done his country much service. Since the election, these victors growl at their opponents like a dog with a bone. What the Vice-President has done to merit a fresh vial of wrath we know not.