Absolutely, so long as the amount in controversy
doesn't exceed the small claims limit in your jurisdiction.
If it does, off to regular court.
And the nice thing about small claims court, if you meet the maximum
limit of course, is that large companies often are lazy about dealing
with the claims. If you are smaller than them, judges will often grant
you a judgement when the big company doesn't show. At that point, with a
court judgement in hand, it is easier to get the carrier's attention.
SLA disputes are usually handled in the sales department by people whose
paycheck is at least partly determined by sales quotas. Paying you the
SLA hurts their paycheck. But a court judgement is usually handled by
the legal department whose paycheck is 100% salary plus bonus for
performance, which in the case of a lawyer would have to do with
settling cases so that they do minimum damage to the company. In the
case of a small claims court judgement, it is cheaper for them to pay
you than to dispute the judgement.
Of course, if you understand the supplier's internal issues from the
beginning, then you can usually escalate it to the right people to
resolve the problem before it gets to court. For instance, sales
management usually have the bonus/quota system rigged so that they can
pay you your SLA but not get hurt in the paycheck.
Don't bang your head against the wall. If the supplier doesn't deal with
issues promptly, work your way through different people in the
organization until you find someone who can act to fix the problem.