In John's June 1998 interview, he says the Barnhills kept the dog often because they had become more attached to it than the children:
3 LOU SMIT: But didn't you board your dog over
4 at the Barnhill's?
5 JOHN RAMSEY: We did. They became attached
6 to him. We had them keep him once or twice and
7 they really loved him. They lost their dog and he
8 died and so they really became attached to
9 Jacques, and it was kind of a good setup because
10 they took care of him and the kids could play with
11 him. So it was, for us, it was, as parents, it was
12 kind of neat.
13 LOU SMIT: So did the kids go over there
14 quite often?
15 JOHN RAMSEY: Yeah. It was not uncommon for
16 them to go over there or Jacques would come over.
17 He'd stay with us sometimes and he became
18 increasingly their dog because they were just so
19 attached to it. And we thought it was kind of nice
20 because they were older. It was nice companionship
21 for them. It was kind of neat for us because the
22 kids could still have a dog and we didn't have to
23 deal with a dog 24 hours a day.
24 So, more and more as time went on, he stayed more
25 at the Barnhills than he did at our house.
Patsy, however, said in her interviews that Jacques was only with the Barnhills when they went away. She was around more and would know better, but at the same time with their travelling, having three different homes, and JBR's pageant schedule, they were gone quite a bit. She confirms JR's statement that the Barnhills had at least one little dog that passed away, and that contributed to them allowing their dog to stay with them.
From Patsy's April 1997 interview:
TT: How often do you talk to the Barnhills or have you talked to the Barnhills in the past?
PR: Uh, I kind of keep, you know their [sic] elderly . . .
TT: Um hum.
PR: . . .and I would kind of, you know every two or three days probably . . .
TT: Kind of keep in touch with them:
PR: Uh huh. They, they kept our dog a lot . . .
TT: Um hum.
[FONT=&]PR: . . .when we went out of town or whatever.
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From Patsy's June 1998 interview:
1 THOMAS HANEY: You say you got a
2 dog. You had a dog up here, right?
3 PATSY RAMSEY: We have a dog in
4 Boulder that we bought.
5 THOMAS HANEY: And what -- what
6 happened, I guess the dog was somewhere else?
7 PATSY RAMSEY: He -- he [sic] had to have
8 a Bichons Frise.
9 THOMAS HANEY: I am sorry?
10 PATSY RAMSEY: A little Bichons
11 thing, white-haired, puffy, fluffy dog. Not a
12 hound dog. It's a little -- I mean, JonBenet --
13 we went to the puppy shop one day and she said,
14 "Oh, that one." And, you know, they put this
15 little white fluffy thing in her arms and, you
16 know, I said wrap it up. Yes, we will take one
17 of everything that goes with it. I mean, it's
18 just putty. It was a darling little dog.
19 He had had [sic] him running across the
20 street to the Barnhills, that was the couple
21 that lived across the street. And they had had
22 a couple of small dogs that they had to put to
23 sleep, so they became very found of Jacques. So
24 when we would go out of town for any length of
25 time, they would dog-sit for us.
In this context, IMO this seems pretty reasonable and not unheard of, although certainly not the norm. Perhaps, too, JBR's infatuation with her doggie waned over time and it wasn't a big deal to her. Then it was a win-win-win (the Barnhills got another little dog they could dote over and didn't have to pay for, JBR got to visit her doggie as she pleased, and the Ramseys didn't have to take constant care of it).
Patsy was questioned somewhat extensively about home security and how a barking dog could have alerted them to an intruder. To this she expressed regret that their dog wasn't there that night. To me, how often the dog was or wasn't home is only really relevant if the intruder wasn't someone known to them who had been sneaking into their house or stalking them on prior occasions. If that's the angle the Ramseys were getting at, then it would make sense to say the dog was rarely at their place when they were home (John implies this, Patsy does not), giving plenty of opportunity for stalking and sneaking without a noisy dog interrupting. I don't know if the Barnhills were interviewed about this and, if they were, whether their story is contradictory.
Otherwise, the reason for having the dog at the neighbors that night is completely legitimate (I've done the same), and doesn't raise suspicions. No reason to lie there.