To be sure, this is a show and probably may things happen as screenwriting goes. And one reason we can conclude from the show is that Horik is cunning - why would he lose just his man against Borg if he can add Ragnar's too, reduce some of his own loses and increase chance for victory?
But as historical possibilities, it is very possible and actually very likely. Being a king doesn't grant you powers, more likely you had to balance among arrogant and ambitious vassals. They do not respond by bringing all their forces to king's disposal just because of royal request. Active troops cost lot of money, and is usually done by some overlapping interest (defense against Heathens, attack with agreed loot division etc). Why would you as Duke/Jarl spent your money and manpower, which may leave you very weak and open to attack, just because some king tells you something?
To be sure, if king has so much wealth and soldiers that even several vassal together can't oppose him, you get very dangerous conqueror around. But that usually doesn't happen, and even if it does it is still not worth losing man to make all vassals obey you unquestionably. King was usually among more powerful of the Dukes/Jarls, but no way that meant he had vast number of soldiers at his disposal. His trouble with Jarl Borg were over resource. Other Jarl's must be promised something juicy to join the war, which Horik doesn't want to give since he wants something juicy from Borg, not to give it away to somebody else. And other Jarls don't really care if Borg wins and becomes new king, it is same for them. Additionally, whoever wins might be weaker and therefore more open to bow to pressures from other Jarls.
To have more soldiers, a king had to make deals with his vassals. Marriage was seen as definite alliance, other deals would have to be made of all sorts. Good and powerful king was always either good in this or had good advisers for this. Horik can't give money to somebody because he fights for money, but getting Ragnars's forces costs only promise that he will join him in raid on England, something Horik already wanted to do anyway. So he gets a powerful Jarl at his side for free.
In addition to all this, at the time in Scandinavia things were even more decentralized than in Europe/Britain. Invasion was the first thing to bring some unity. Being a king probably meant that this person became renowned in raiding. It is not likely that any Jarls actually paid any taxes and probably acted completely independently. This naturally changed a lot in next few centuries. But that early and without fast means of transportation, ruling over other was very hard and impractical, which is why smaller decentralized units (county, duchy etc) existed in the first place. As roads, wagons and ships improved, so did the efficiency of the supreme ruler. Nowadays we have centralized countries with one government, although we still have separate bodies to govern the cities and counties. Back in that time there is no real country, just many small units that might or might not follow one ruler.