Jim Munson has one of those refugee stories that warm your heart. He belongs to a private group in Ottawa that is sponsoring a Syrian family. Last winter, he was asked to make sure the boys got skates and hockey sticks. So he found them skates and hockey sticks and took them skating on the canal. Their first words in English were, "He shoots, he scores!"

If only it were all that easy.

Canada is held up as a model to the world for our warm welcome to Syrian refugees. People marvel that Canadians are clamouring to sponsor refugee families, sight unseen. In Canada, people complain not that we're taking in too many refugees, but that we're taking in too few, and eager sponsors are still waiting.

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We are good people. But it's a bit too soon for self-congratulation. Across Canada, refugees have been turning to food banks because they can't make ends meet. Toronto's Daily Bread Food Bank found that Syrian families have less than $400 a month left after they pay the rent. In Montreal, about 2,000 refugees depend on Moisson Montreal, the country's largest food bank. In Winnipeg, Yasmin Ali, who runs the city's newest food bank, told the CBC, "This is very stressful for them because I guess they didn't expect this."

To be fair, the federal government is in a bind. Its subsidies are pegged to provincial welfare rates. If refugees were to get better treatment than Canadians, Canadians would get cranky.

Canada has a robust network of social-service agencies. But none were prepared for the deluge, and they didn't get more funding to deal with it. In 2014, Canada resettled about 13,500 refugees; this year the number, from all countries, is expected to be more than 50,000. Language training, translators, trauma and mental-health care – all are in short supply. Refugees can't find jobs until they learn English (or French in Quebec), but some language schools have run out of money to meet the demand.

"We need more than love," Mr. Munson, who heads the Senate standing committee on human rights, told me. His committee has been monitoring the refugee program. To say the least, it has a lot of growing pains. In his view, we should be cautious about exporting the Canadian model while we figure out how to address them.

On top of that, the rate of processing new refugees has slowed to a crawl. As hopeful families live in limbo abroad, sponsorship groups are increasingly fed up with an unresponsive immigration bureaucracy.

John Bryan belongs to a Toronto group that has agreed to sponsor a Syrian family currently living in Saudi Arabia. They've formed strong bonds. They've even arranged for one of the children, an engineering student, to go to York University. Now they have been abruptly informed that the family won't be processed for three years or more. Why? Perhaps lack of resources, or perhaps bureaucratic logistics. Whatever the case, he told me, "The government made a commitment and they're not going to keep it."

Privately sponsored refugees do better than government-sponsored ones, in part because they have a better web of support. But for both groups, the money runs out after a year. Then what happens?

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The answer depends on employability. And the early signs aren't very good. "Basically I can't refer them to any employer because they don't have basic communication skills," one settlement worker told the Huffington Post. Also, few Syrian refugees are highly educated. (One survey of refugees in Hamilton found that two-thirds of those aged 15 and older had a high-school education or less.) And in some parts of Canada, especially Alberta, the job market is terrible.

"The last thing anybody wants to see is people moving from one government program to a provincial welfare roll," Mr. Munson says. But it's likely that more than a few will.

No one expects the newcomers to be self-sufficient right away. That's not why we took them in. We took them in because we have a moral obligation to ease the suffering of people who have survived an unspeakable humanitarian catastrophe. We owe it to them – and to ourselves – to give them the best support we can, and also to be realistic about what we can take on and what we can expect.

It would be foolish to open the doors even wider (as some people think we should) while so many newcomers are struggling to find purchase. It would also be smart of us to remember that integration takes more than a generation, and that not all immigrant groups are equally successful, no matter how much help they get.

The Vietnamese boat people thrived in Canada. How will the Syrians fare? We'll find out in 20 or 30 years. Meantime, we shouldn't underestimate the challenge.