The limit occured when in the winter/spring of 1944 the boat obstacles were extended across all the beaches & to the low water line. A factor that did not exist in 1943 outside the port defenses.
Standard doctrine for beach crossing is to land the first wave at high tide, this provides the shortest amount of exposed beach for the first assault wave to cross. There are some disadvantages for the navy, but generaly its prefered to land the initial force at high tide. The appearance of the thick bands of boat obstacles and mines made it impractical to beach the landing craft at high tide. Both the US & Brits spent a lot of effort from December 1943 through February to find a way to remove the obstacles in advance of the landing craft. The USN built a replica of the Normandy boat obstacles on a Florida beach in January 1944 & tried things like firing torpedos at the beach, rocket carried line charges from boats, large caliber naval guns, massed rockets fired from boats, air bombardment. Those & other ideas did not do the task. The only reliable method was to flood the first wave with demolition teams who would destroy the obstacles with explosive charges.
So the first wave was landed at low tide to prevent the boats from being wrecked on the obstacles, and this left the obstacles momentarily dry allowing the demo teams to work faster.
The time of day is a different matter. The low/high tide cycle runs on a fourteen day pattern, with the tide line occuring at sucessivly different times of day. For several reasons the planners (21 Army Group had primary responsibility in this) wanted the assault to start at earliest light (Begaining Nautical Twilight in the technical language) That meant the preparatory fires from the naval guns would start at first light & then the first assault wave would roll in at the low tide hour shortly after. On June 6th BNT came at a little after 05:00 & the naval/air attacks started shortly after as the thick haze cleared. The low tide time came first on UTAH Beach around 06:15-06:20, on OMAHA Beach around 06:35, & sucessively later on the British beaches. In the latter cases the mudflats or shoals in front of the beaches required a delay so there would be enough water to float the landing craft over the shoals. That reduced the time for the demo teams to clear boat lanes in the obstacles.
Few beach landings are executed at dawn as @ Normandy. Op AVALANCHE at Anzio was executed at 01:30 six or seven hours before dawn, when the high tide occured. This had to do with the desire to execute the operation as soon as possible. Op AVALANCHE at Salerno was also started hours before dawn on the high tide. Op GALVANIC against Betio Island had the first wave go in on the early afternoon high tide. For most operations the question of date comes first and hour of the tide is secondary. For Op NEPTUNE @ Normandy several factors came together to make a dawn assault at low tide the choice. That limited the dates to the 8/8/10 or 22/23/24 May, and the 5th/6th/7th & 19/20/21 June.
Had the planners of Op NEPTUNE prefered to match the beach assault to the airborne landing that started at 01:30 then they would have selected dates with low tide at approx that hour. Or they might have made other choices. Bottom line is that on the Normandy beaches the anti boat obstacles meant they would be landing at low tide whatever day they choose. In 1943 the anti boat obstacles would not be a factor.
Rear Adm Ellesbergs book 'The Far Shore' has a very readable & professional explanation of this. Any number of other histories of Op NEPTUNE & OVERLORD also describe the influences on the selection of date and hour.