First make sure the lowest suite category is available for all days through out stay. You may want to break up your stay into multiple one night stays if you are willing to switch rooms.

When checking in, it is very possible that all suites are being occupied or checked out but in "dirty" status (not cleaned yet) at that time. When this is the case, front desk agents tend to be lazy and tell you that they are fully booked etc. and try assign you to a room in your originally booked room type.

If checking in before 2pm, 3pm etc [depending on property] (or official check-out time), ask the front desk agent to check the "departures list" for that day to see if any suites are due to check out that day and whether any of them have late check out. Be nice and tell them that you are willing to wait until one of the guests check out (which could even be well after 4PM for those already occupied and have late check out). The key is that if you are nice enough, they might even pre-assign you to that room and it will be yours once it is cleaned.

If checking in after official check-in time (but before late check-out time of 4PM), ask the front desk agent if there are any suites that are currently in "dirty" status (eg. checked-out but not cleaned yet). Also ask the front desk agent to check the "departures list" to see if anyone has a 4PM check-out. If so, tell them you are willing to wait.

If checking in after 4PM, ask if there are still suites that are in "dirty" status and tell them that you are willing to wait.

Also, ask about whether there any suites are currently in "out of service" status. Some of the higher end hotels block off entire floors (usually where most rooms are suites or regular rooms that can be connected to suites) in case a VIP shows up. If needed, they can quickly put one of those suites into service if a paying guest books. Note: It is possible that some hotels will still take bookings for suites when in reality they are in "out of service" status in the hotel's property management system.

When asking about "departure lists", "dirty status", "out of service" etc, it is common for most front desk agents to have to go into the back room and ask perhaps the front office manager who has better access to the property management system.