According to NFL.com, the Patriots defense is the 4th worst in the league on 3rd down, better than only the Colts, Chiefs, and Bengals. I have to imagine the Patriots are the worst in 3rd and long situations.
Looking up the games and finding their 3rd and double-digit yards situations, this is what I found.
Positive outcome Negative outcome
Week 1
4Q
3-13-KC 17 (10:12) (Shotgun) 11-D.Huard pass short right to 80-J.Webb to KC 23 for 6 yards (29-L.Sanders).
4-7-KC 23 (9:37) 2-D.Colquitt punts 41 yards to NE 36, Center-51-J.Darche. 21-D.O'Neal to NE 38 for 2 yards (52-E.Walden, 51-J.Darche).
Week 2
2Q
3-18-NYJ 12 (3:07) (Shotgun) 4-B.Favre pass short middle to 29-L.Washington to NYJ 19 for 7 yards (37-R.Harrison).
4-11-NYJ 19 (2:21) 7-B.Graham punts 29 yards to NYJ 48, Center-85-J.Dearth, out of bounds.
3Q
3-22-NYJ 11 (11:30) (Shotgun) 4-B.Favre pass deep left to 83-C.Stuckey to NYJ 39 for 28 yards (31-B.Meriweather).
4Q
3-23-NYJ 7 (5:18) (Shotgun) 4-B.Favre pass incomplete deep left to 87-L.Coles (27-E.Hobbs).
4-23-NYJ 7 (3:57) 7-B.Graham punts 55 yards to NE 38, Center-85-J.Dearth. 33-K.Faulk to NYJ 47 for 15 yards (16-B.Smith, 15-W.Wright).
Week 3
N/A
Week 4
Bye
Week 5
4Q
3-11-SF 44 (1:34) 14-J.O'Sullivan pass incomplete deep left to 83-A.Battle.
4-11-SF 44 (1:30) 14-J.O'Sullivan pass incomplete short right to 88-I.Bruce.
Week 6
2Q
3-11-SD 48 (4:55) (Shotgun) 17-P.Rivers pass incomplete deep left to 83-V.Jackson (31-B.Meriweather). NE-31-B.Meriweather was injured during the play.
4-11-SD 48 (4:48) 5-M.Scifres punts 46 yards to NE 6, Center-50-D.Binn, downed by SD-81-K.Osgood.
3Q
3-10-NE 26 (4:22) (Shotgun) 17-P.Rivers pass short left to 80-M.Floyd to NE 17 for 9 yards (27-E.Hobbs).
4-1-NE 17 (4:22) 10-N.Kaeding 35 yard field goal is GOOD, Center-50-D.Binn, Holder-5-M.Scifres.
Week 7
2Q
3-10-DEN 10 (10:38) (Shotgun) 6-J.Cutler scrambles left end to DEN 17 for 7 yards (36-J.Sanders, 90-L.Smith). Penalty on DEN, Illegal Formation, declined.
4-3-DEN 17 (10:20) 1-B.Kern punts 50 yards to NE 33, Center-83-M.Leach. 83-W.Welker ran ob at NE 44 for 11 yards.
3-10-DEN 20 (6:54) (Shotgun) 6-J.Cutler pass short right to 19-E.Royal to DEN 33 for 13 yards (37-R.Harrison). NE-58-P.Woods was injured during the play.
3Q
3-14-DEN 8 (9:40) (Shotgun) 28-M.Pittman left guard to DEN 16 for 8 yards (59-G.Guyton, 37-R.Harrison).
4-6-DEN 16 (9:03) 1-B.Kern punts 56 yards to NE 28, Center-83-M.Leach. 83-W.Welker pushed ob at DEN 28 for 44 yards (83-M.Leach).
4Q
3-10-DEN 28 (7:56) (Shotgun) 6-J.Cutler pass short right to 15-B.Marshall ran ob at DEN 40 for 12 yards. PENALTY on DEN-28-M.Pittman, Chop Block, 14 yards, enforced at DEN 28 - No Play.
3-24-DEN 14 (7:25) (Shotgun) 6-J.Cutler pass short left to 89-D.Graham pushed ob at DEN 35 for 21 yards (35-M.Richardson). PENALTY on DEN-82-D.Jackson, Offensive Holding, 10 yards, enforced at DEN 31.
3-17-DEN 21 (6:54) 6-J.Cutler sacked at DEN 15 for -6 yards (93-R.Seymour).
4-23-DEN 15 (6:35) 1-B.Kern punts 45 yards to NE 40, Center-83-M.Leach. 83-W.Welker to NE 48 for 8 yards (52-L.Green).
Week 8
2Q
3-10-STL 20 (11:16) (Shotgun) 10-M.Bulger pass deep middle to 14-K.Burton to STL 47 for 27 yards (31-B.Meriweather).
3-25-NE 38 (7:36) (Shotgun) 10-M.Bulger sacked at NE 40 for -2 yards (93-R.Seymour).
4-27-NE 40 (7:12) 5-D.Jones punts 40 yards to end zone, Center-45-C.Massey, Touchback.
3Q
3-14-NE 37 (8:32) (Shotgun) 10-M.Bulger pass short left to 81-T.Holt pushed ob at NE 26 for 11 yards (24-J.Wilhite, 36-J.Sanders). STL 81-Holt now has 12,153 career receiving yards, passing Charlie Joiner for 15th all-time.
4-3-NE 26 (8:08) 3-J.Brown 44 yard field goal is GOOD, Center-45-C.Massey, Holder-5-D.Jones.
4Q
3-15-STL 15 (7:45) (Shotgun) PENALTY on STL-70-A.Barron, False Start, 5 yards, enforced at STL 15 - No Play.
3-20-STL 10 (7:45) (Shotgun) 10-M.Bulger pass short left to 82-D.Hall to STL 15 for 5 yards (24-J.Wilhite).
4-15-STL 15 (7:12) 5-D.Jones punts 46 yards to NE 39, Center-45-C.Massey. 83-W.Welker to NE 47 for 8 yards (52-C.Draft, 55-G.Stills).
3-10-STL 37 (2:18) (Shotgun) 10-M.Bulger pass short middle to 82-D.Hall to 50 for 13 yards (96-A.Thomas).
3-10-NE 33 (1:26) (Shotgun) PENALTY on STL-68-R.Incognito, False Start, 5 yards, enforced at NE 33 - No Play.
3-15-NE 38 (1:26) (Shotgun) 10-M.Bulger pass deep right intended for 14-K.Burton INTERCEPTED by 21-D.O'Neal at NE 16. 21-D.O'Neal to STL 37 for 47 yards (10-M.Bulger).
Week 9
2Q
3-10-IND 48 19) (Shotgun) 18-P.Manning pass deep right to 88-M.Harrison to NE 26 for 26 yards (36-J.Sanders).
Week 10
2Q
3-10-NE 46 (1:52) (Shotgun) 5-T.Edwards pass short middle to 81-J.Hardy to NE 39 for 7 yards (23-J.Webster, 58-P.Woods).
Timeout #1 by BUF at 01:04.
4-3-NE 39 (1:04) 8-B.Moorman punts 36 yards to NE 3, Center-72-R.Neill, downed by BUF-17-J.Jenkins.
3Q
3-12-BUF 17 (2:40) (Shotgun) 5-T.Edwards pass short right to 83-L.Evans ran ob at BUF 32 for 15 yards.
4Q
3-11-NE 42 (11:18) (Shotgun) 5-T.Edwards pass incomplete deep right to 84-R.Royal (51-J.Mayo).
4-11-NE 42 (11:11) 8-B.Moorman punts 34 yards to NE 8, Center-72-R.Neill, fair catch by 33-K.Faulk.
Week 11
2Q
3-9-NYJ 31 (1:56) PENALTY on NYJ-4-B.Favre, Delay of Game, 5 yards, enforced at NYJ 31 - No Play.
3-14-NYJ 26 (1:56) 29-L.Washington right guard to NYJ 33 for 7 yards (93-R.Seymour, 59-G.Guyton).
Timeout #2 by NE at 01:51.
4-7-NYJ 33 (1:51) 6-R.Hodges punts 35 yards to NE 32, Center-85-J.Dearth, fair catch by 33-K.Faulk.
3Q
3-14-NYJ 29 (11:45) (Shotgun) 4-B.Favre sacked at NYJ 20 for -9 yards (93-R.Seymour).
4-23-NYJ 20 (11:26) 6-R.Hodges punts 46 yards to NE 34, Center-85-J.Dearth. 83-W.Welker to NE 36 for 2 yards (31-A.Carroll, 16-B.Smith).
OT
3-15-NYJ 15 (14:19) (Shotgun) 4-B.Favre pass short middle to 81-D.Keller to NYJ 31 for 16 yards (31-B.Meriweather, 51-J.Mayo).
Week 12
1Q
3-14-MIA 25 (9:17) 10-C.Pennington pass short middle to 34-R.Williams to MIA 31 for 6 yards (50-M.Vrabel).
4-8-MIA 31 (8:45) (Punt formation) 2-B.Fields punts 52 yards to NE 17, Center-92-J.Denney. 83-W.Welker to NE 25 for 8 yards (56-C.Anderson).
2Q
3-12-MIA 41 (9:36) (Shotgun) 10-C.Pennington pass deep right to 83-G.Camarillo to NE 39 for 20 yards (21-D.O'Neal)
3-10-NE 49 16) (Shotgun) PENALTY on MIA-19-T.Ginn, False Start, 5 yards, enforced at NE 49 - No Play.
Timeout #2 by NE at 00:16.
3-15-MIA 46 16) (Shotgun) 10-C.Pennington sacked at MIA 39 for -7 yards (50-M.Vrabel).
3Q
3-20-NE 37 (1:50) (Shotgun) 10-C.Pennington sacked at NE 41 for -4 yards (93-R.Seymour). Penalty on MIA-65-J.Smiley, Offensive Holding, declined.
4-24-NE 41 (1:37) (Punt formation) 2-B.Fields punts 28 yards to NE 13, Center-92-J.Denney, fair catch by 83-W.Welker.
4Q
3-10-MIA 24 (6:02) 10-C.Pennington pass short middle to 23-R.Brown to MIA 34 for 10 yards (59-G.Guyton). 1st Down Measurement.
3-10-MIA 44 (4:15) (Shotgun) 10-C.Pennington pass incomplete short right to 23-R.Brown.
4-10-MIA 44 (4:10) (Shotgun) 10-C.Pennington pass incomplete deep right to 15-D.Bess [93-R.Seymour]. Penalty on MIA-34-R.Williams, Chop Block, declined. Penalty on MIA-65-J.Smiley, Chop Block, declined.
Week 13
1Q
3-12-NE 27 (5:42) (Shotgun) 7-B.Roethlisberger pass deep left to 85-N.Washington to NE 11 for 16 yards (27-E.Hobbs, 31-B.Meriweather).
2Q
3-10-PIT 13 (11:57) (Shotgun) 7-B.Roethlisberger pass incomplete short left to 10-S.Holmes (27-E.Hobbs).
4-10-PIT 13 (11:52) 17-M.Berger punts 38 yards to NE 49, Center-61-J.Retkofsky. 83-W.Welker to PIT 44 for 7 yards (61-J.Retkofsky, 92-J.Harrison). PENALTY on NE-28-A.Spann, Illegal Block Above the Waist, 10 yards, enforced at 50.
I would imagine this would be much more damning if "3rd and long" included situations like 3rd and 6, 3rd and 7, and so forth.