Fixed simply by removing a sneaky execute hiding after the procedure.

When I execute my stored procedure, it keeps on going in an infinite loop when I only want it to iterate once.

It successfully inserts the correct number of rows into my PartitionInfo table, but when I use the cursor to iterate through, it will keep going until I get an error:

Msg 217, Level 16, State 1, Procedure ConnectorFile.GetPartitionInfo, Line 85 [Batch Start Line 84]
Maximum stored procedure, function, trigger, or view nesting level exceeded (limit 32)

This is my code:

CREATE OR ALTER PROCEDURE [ConnectorFile].[GetPartitionInfo]
    @tableName NVARCHAR(128)
AS
BEGIN
    SET NOCOUNT ON;

    -- Declare a table variable to hold the partition information
    DECLARE @PartitionInfo TABLE 
                           (
                               table_name NVARCHAR(128),
                               partition_range NVARCHAR(128),
                               file_group_name NVARCHAR(128),
                               partition_number INT,
                               number_of_rows INT
                           );

  -- Retrieve partition information and insert it into the table variable
  INSERT INTO @PartitionInfo (table_name, partition_range, file_group_name, partition_number, number_of_rows)
      SELECT DISTINCT
          o.name AS table_name,
          CONVERT(NVARCHAR(128), rv.value) AS partition_range,
          fg.name AS file_group_name,
          p.partition_number,
          p.rows AS number_of_rows
      FROM
          sys.partitions p
      INNER JOIN
          sys.indexes i ON p.object_id = i.object_id 
                        AND p.index_id = i.index_id
      INNER JOIN
          sys.objects o ON p.object_id = o.object_id
      INNER JOIN
          sys.system_internals_allocation_units au ON p.partition_id = au.container_id
      INNER JOIN
          sys.partition_schemes ps ON ps.data_space_id = i.data_space_id
      INNER JOIN
          sys.partition_functions f ON f.function_id = ps.function_id
      INNER JOIN
          sys.destination_data_spaces dds ON dds.partition_scheme_id = ps.data_space_id 
                                          AND dds.destination_id = p.partition_number
      INNER JOIN
          sys.filegroups fg ON dds.data_space_id = fg.data_space_id
      LEFT OUTER JOIN
          sys.partition_range_values rv ON f.function_id = rv.function_id  
                                        AND p.partition_number = rv.boundary_id
      WHERE
          o.object_id = OBJECT_ID(@tableName);

    DECLARE @table_name NVARCHAR(128);
    DECLARE @partition_range NVARCHAR(128);
    DECLARE @file_group_name NVARCHAR(128);
    DECLARE @partition_number INT;
    DECLARE @number_of_rows INT;

    -- Cursor to iterate through each row in the table variable
    DECLARE cur CURSOR FOR 
        SELECT table_name, partition_range, file_group_name, partition_number, number_of_rows
        FROM @PartitionInfo
        ORDER BY partition_number;

    OPEN cur;

    FETCH NEXT FROM cur INTO @table_name, @partition_range, @file_group_name, @partition_number, @number_of_rows;

    WHILE @@FETCH_STATUS = 0
    BEGIN
        IF @number_of_rows > 5000000
        BEGIN
            PRINT CONCAT('Splitting partition: ', @partition_number, ', Rows: ', @number_of_rows);
        END
        ELSE
        BEGIN
            PRINT CONCAT('NOT Splitting partition: ', @partition_number, ', Rows: ', @number_of_rows);
        END

        FETCH NEXT FROM cur INTO @table_name, @partition_range, @file_group_name, @partition_number, @number_of_rows;
    END;

    CLOSE cur;
    DEALLOCATE cur;
END;

As said, I have checked if PartitionInfo table, it contains the correct number of rows (in my case 8), and 8 exists, but when it has executed 8 it just starts over!