After being put into bat the defending champions made 257 for 8 off 43 overs, Ben Wright (82), Carl Wilson (53), Jordan Temple (41) and Peter Morgan (40) doing the business with the bat.
Zain Abbas took three Bourne wickets.
Grantham responded with 264 for 9 after being given a revised target of 266.
Joe Kendall (85), Abbas (56) and Dhruv Shahi (38) top scored.
Brandon Diplock and Robert Bentley both took three wickets apiece as Bourne won by one wicket.
Second-place Sleaford's home contest with Scunthorpe Town was cancelled.
Lindum, in fourth, left Nettleham with a 22-run success.
Robert Cook (58 no), Pranav Pothula (58) and Will Taylor (56) helped the visitors to 234 for 9.
In response, Mitchell Slattery (47) and Rhett Bridgens (37), helped Nettleham to 176 for 7, after being given a revised target of 199.
Nettleham's Bridgens took three wickets.
There were some impressive batting scores as Louth left Skegness with an eight-wicket success.
Jamie Epton's 88 helped the hosts post 176 for 5.
Louth-with a revised target of 196, reached 198 for 2 thanks to Samueal Jones (88 no) and Remembrance Nyathi (78 no).
Woodhall Spa secured a 53-run success against Spalding at Jubilee Park.
Tom Caswell (28 ret), Oliver caswell (21), Prasanna Jayawardene (21) and Gareth Grant (20) helped the Spa men to 156 for 7.
Alex King and Tom White both took three wickets as Spalding were out for 103, dsespite the best efforts of Josh Lawrence (38).
Market Deeping earned a 194-run success over Bracebridge Heath.
Zeeshan Manzoor (84), Josh Smith (55), Ashar Hafeez (51) and Sam Malton (42) led Deeping to 276 before they were dismissed, Joseph Mansford taking five wickets for the loss of 49 runs.
Tom Mcilwaine top scored for Bacebridge with 23.
Hafeez (5-18) and Hayatullah Niazi (3-15) did the damage with the ball.
Lincs ECB Premier
Bourne 257-8, Grantham 264-9; Market Deeping 276 all out, Bracebridge Heath 82 all out; Nettleham 176-7, Lindum 234-9; Skegness 176-5, Louth 198-2; Woodhall Spa 156-7, Spalding 103 all out.
South Lincs and Border League
Premier Division: Boston 50-3, Market Deeping 2nds 118-9 - abandoned; Freiston L&L 188 all out, Woodhall Spa 2nds 188 all out; Grantham 2nds v Bourne 2nds - abandoned; Heckington v Moulton Harrox - abandoned; Long Sutton v Billingborough 55-3 - abandoned; Newborough 173-9, Sleaford 2nds 181 all out.
Division One: Ancaster v Boston 2nds - cancelled; Moulton Harrox 2nds 104-3, Baston 100 all out; Orton Park 2nds 39-1 v Grantham 3rds - cancelled.
Division Two: Belton Park v Skegness 2nds - abandoned; Orton Park 3rds 56 all out, Long Sutton 2nds 194-9; Spalding 3rds 138-9, Boston 3rds 103 all out.
Lincolnshire County League
Premier Division: Grimsby Town 130 all out, Haxey 255-7; Holton-Le-Clay 212-8, Caistor 79 all out; Lindum 2nds 95 all out, Outcasts 98-6; Market Rasen 185 all out, Scothern 62-1 - abandoned; Scunthorpe Town 2nds 158 all out, Cleethorpes 2nds 207-9.
First Division: Alkborough 124-6, Bracebridge Heath 2nds 123 all out; Appleby Frodingham 2nds 212-7, Broughton 181 all out; Barton Town 167 all out, Hartsholme 216-6; Clee Town Laportes 186-9, Cherry Willingham 188-9.
Second Division: Cleethorpes 3rds 243-7, Holton-Le-Clay 2nds 246-4; Hartsholme 2nds 60-7, Grimsby Town 2nds 127 all out - abandoned; Hibaldstow 165 all out, Market Rasen 2nds 198-5; Lindum 3rds 219-1, Nettleham 2nds 65-2; Normanby Park 86-2, Alford & District 84 all out.
Third Division East: Cherry Willingham 2nds 120-3, Nettleham 3rds 118-8.
Third Division West: Broughton 2nds 88-0, Messingham 2nds 85 all out.
Lincoln and District League
Premier Division: Lindum 126-4, Sleaford 124 all out; Owmby 125-8, Bracebridge Heath 171 all out; Woodhall Spa 186-4, Ancaster 185-8.
Division One: Barkston & Syston 220-6, Hartsholme 219-6, Caythorpe 105-0, Blankney 104 all out; Cherry Willingham 187-6, Morton 180 all out; Nocton 105 all out, Billingborough 50 all out.
Division Two: Heckington 218-6, North Scarle 163-7; Market Rasen 75-2, Bassingham 73 all out; Old Lincolnians 169 all out, Lindum 2nds 79 all out.
Division Three: Barrowby v Revesby 2nds - cancelled; Nettleham Academy 130-4, Reepham 129 all out; Rustons v Owmby Trojans - Owmby conceded.
East Lindsey League
Cleethorpes 97-6, Alford & District 96 all out; Grimoldby & District 140-2, Louth Taverners 139 all out; Grimsby Town Development 138-7, Clee Town Laportes 134 all out; South Bank 103 all out, Barton Town 189-9.