That varies from hosting provider to hosting provider.
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Some will suspend your account if you go over your disk space limit or monthly bandwidth quota, while others will let you off with a warning.
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You should be aware, though, that some companies will charge you expensive overage fees for exceeding their limits. When that happens, you may have a nasty surprise waiting for you when you open next month's bill.
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When in doubt, you should always carefully read over the Terms of Service agreement, as well as any other legal agreements offered to you at signup.
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If you see that you are about to go over your space or bandwidth limits, you should immediately contact your hosting provider and ask about upgrading to a plan with higher limits.