Town were forced to settle for just one point from a goalless draw with Carlisle United at Brunton Park.
In a game between two of League Two's bottom five, the Mariners created several good chances and would have been disappointed not to win.
Harry Davis had an early goal disallowed for offside and James Hanson hit the post with a header as Town dominated the early stages, before a deflected Robson free kick hit the bar and substitute Charles Vernam also had an opportunity just before half-time.
The visitors continued to be the better team after the break but were unable to take their chances, although they remained solid at the back to ensure they would take some reward from the match.
Three changes were made to the Town starting line-up after last week's defeat by league leaders Swindon Town, with Luke Hendrie returning in place of the suspended Elliott Hewitt as Ethan Robson and Max Wright were preferred to Elliott Whitehouse and Vernam.
James McKeown started in goal, with Hendrie, Davis, Luke Waterfall and Liam Gibson forming the defence.
Harry Clifton, Jake Hessenthaler and Ethan Robson were the midfielders, with Matt Green and Wright either side of Hanson in attack.
Whitehouse and Vernam were both named on the bench, along with Sam Russell, Mattie Pollock, Ludvig Ohman, Ahkeem Rose and Harry Cardwell.
Carlisle were one of only four clubs below Grimsby in the league table at kick-off and were on a similar run of form, meaning the game took on extra significance for both teams as they looked to put extra distance between them and the bottom.
The Mariners thought they had taken the lead on four minutes when a free kick from Hessenthaler on the right was headed into the net by Davis but the goal was ruled out for offside against the centre-back.
And the visitors' bright start continued when Hanson had a chance to open the scoring on seven minutes, although his header bounced off the bottom of the post and rolled away from goal.
A low cross from Green was cleared, before Robson became the first to be shown a yellow card after a poor challenge on Carlisle's Harry McKirdy near the halfway line.
McKirdy was involved again soon afterwards, setting off on a driving run through the middle which took him into the Town penalty area before he was eventually stopped by a good tackle from Waterfall.
The hosts won a corner when Hallam Hope cut in from the left and saw a shot deflected wide. The ball from the flag was met by Jarrad Branthwaite but he could only send it wide.
A Carlisle free kick from the right on 23 minutes was cleared, before an effort from Hope came back off the bar and the ball was cleared.
Town were inches away from taking the lead on 27 minutes when a free kick by Robson took a deflection and wrong-footed keeper Adam Collin, only for the ball to bounce back off the bar. Hanson was penalised for a foul as he challenged for the loose ball.
At the other end, Mike Jones fired just past the far post after a Carlisle break, with Nathan Thomas playing the ball back to him on the edge of the box.
A cross from Wright was turned behind for a Town corner on 32 minutes but Hessenthaler's ball from it was headed away by Aaron Hayden.
A home corner then came four minutes later, which was headed clear by Hanson.
Hessenthaler had an opportunity after getting in behind the defence to meet a header by Green, although he could not find the net from a tight angle and the ball was scrambled out for a corner by Collin.
Robson showed determination to break into the box on 42 minutes and fire the ball across, where it was cleared for a corner. From the flag, the ball was met by Waterfall but he could not direct his header on target.
Hanson went down injured after the corner and after receiving treatment, the striker was unable to continue and Vernam was sent on in his place.
The substitute's first action saw him presented with a great chance when defender Canice Carroll's short back-pass fell straight into his path. But Vernam's first touch was too heavy and the hosts survived as the ball went out for a goal kick.
Two added minutes were played at the end of the half and with no further opportunities, the game remained goalless at the break.
Town had played the better football and created a number of good chances but were unable to take any of them.
Christie Elliott replaced Carroll at right-back for the home side before the restart and an early opening came for Town after a mistake by Collin saw the ball won by Hessenthaler at the edge of the Carlisle box. The midfielder got it back from Green but could not pick a team-mate out with his ball to the far post.
Good play by the visitors saw Hendrie keep the ball in on the right to start a passing move which eventually led to a chance for Hessenthaler, although he could only shoot at the keeper.
Hessenthaler was pulled back level with the left corner of the Carlisle penalty area on 59 minutes but Town could not turn the resulting free kick into a goalscoring opportunity.
A run through the middle by Wright launched another attack, as the winger found Hessenthaler on the right but his cross was headed at Collin by Green.
A chance for Carlisle came when a cross from the right fell for McKirdy at the far post but he could only volley harmlessly wide.
McKirdy was replaced by Mo Sagaf for the hosts, before Green was booked after he was adjudged to have blocked a kick by keeper Collin.
A low shot from Hope was vlocked shortly before a home corner, which was taken quickly but ultimately came to nothing as Jones blasted a shot high over the bar from 20 yards.
Clifton did well to get into the penalty area on 69 minutes as he went through two challenges but a bad bounce came just as he looked to cross the ball and he was unable to find a man in black and white.
Town kept attacking and Verman went close to breaking the deadlock but his low effort went just past the far post.
Gibson was booked for a foul on the home right on 76 minutes but the free kick was headed clear by Waterfall.
Rose replaced Green in Town's second change with 13 minutes to go and he was involved in creating their next chance, playing the ball though for Vernam, who hit the side-netting.
Olufela Olomola came on for Stefan Scougall for the home side, before Wright became the latest Town player to be booked when he brought Gethin Jones down near the halfway line.
Wright was replaced by Whitehouse with six minutes left on the clock and a home free kick had to be dealt with soon afterwards, with a good block ensuring the scores remained level.
A foul by Waterfall gave Carlisle another free kick inside the Grimsby half but again, no damage was caused as the referee's whistle blew for a foul when Thomas sent the ball into the box.
Four added minutes were played at the end of the game and an opportunity came for Vernam in the third of them, although he dragged his low shot wide from the edge of the penalty area.
That proved to be the last chance for either side to grab all three points as both were forced to settle for a draw.
Town are next in action on Saturday, when Scunthorpe United are the visitors to Blundell Park.
Carlisle: Collin (C), Carroll, Jones, Hayden, Branthwaite, Jones, Thomas, Scougall, McKirdy, Hope, Loft.
Subs: Gray, Birch, Elliott. Knight-Percival, Sagaf, Olomola, Sorensen.
Town: McKeown (c), Hendrie, Davis, Waterfall, Gibson, Hessenthaler, Robson, Clifton, Green, Hanson, Wright.
Subs: Russell, Pollock, Öhman, Whitehouse, Rose, Cardwell, Vernam.
Referee: Marc Edwards
Attendance: 3,653 (217)
Possession: Carlisle 50% - 50% Town
Next Up: Scunthorpe Utd