Generally, insurance companies are not allowed to have you wait for your coverage to begin; however, if you do have a pre-existing condition, they might be able to have you wait for coverage.
In some cases, your out of pocket costs might be refused by a Medigap insurance company for any preexisting health conditions for a period of time. A six-month period called (pre-existing condition waiting period) after which the pre-existing problem would be covered.
Conditions diagnosed, treated, etc. 6 months prior to the start of your coverage might be excluded. If this is the case the excluded pre-exiting condition will be covered after a 6 month waiting period.
The condition will still be covered when you get Medicare covered services that or Original Medicare still covers that condition regardless if your out of pocket costs are not covered by your Medigap Policy. Coinsurance and copayments are still your responsibility.