Robson is aiming to become the first British woman in 29 years to make it through to the quarter-finals of Wimbledon, but if she is looking for an easy passage to the last eight then she can forget it.
The 19-year-old's refusal to submit to defeat this week has won her many admirers on and off the court.
Robson's form this week suggests she is no longer happy to settle for plucky British battler status. There is an inherent self-belief about her play, and off the court she shows maturity and confidence beyond her years.
Yet Kanepi is clearly cut from the same cloth. She may be 46th in the rankings - 12 places behind Robson - but the Estonian thinks she is so good that she could even go on to win Wimbledon this year.
"If I play every day very well, then it's possible," the 28-year-old from Haapsalu said.
"I feel that I am close to winning a grand slam, but it depends on how I play.
"I played well to win the Brisbane International last year so it's possible."
A more forensic analysis of Kanepi's form suggests she has every reason to feel confident about beating Robson when she steps out on Court One at 1pm on Monday.
Kanepi would probably be near her all-time high ranking of 15th had she not been out for six months because of an Achilles injury earlier this year.
She has also enjoyed success at SW19 before. In 2010 she reached the quarter-finals, becoming only the sixth qualifier in the Open era to make the last eight.
The way Kanepi saw off seventh seed Angelique Kerber, who like Robson is a left-hander, also gives credence to her suggestion that she is playing some of the best tennis of her 13-year professional career.
"I think I am playing my best tennis," Kanepi added.
"I am playing really well. I think my game is more stable than it was three years ago and I think I'm moving better.
"I'm fitter, I'm faster and I'm more aggressive. I'm not afraid to go in anymore."
Kanepi, who has four tour titles to her name, already has one British scalp in the bag after dismissing Doncaster's Tara Moore in the first round.
She then beat last year's semi-finalist Kerber before knocking out American Alison Riske in straight sets yesterday.
Monday's last-16 match will be the first meeting between the two and Kanepi will have done plenty of research on her opponent before they meet.
"I have never played Laura before and I haven't seen any of her games this week," Kanepi said.
"Left-handers are always difficult to play on grass, but I had that against Kerber and won so I'm quite used to it."
If Monday's match is anything like Robson's roller coaster against Marina Erakovic, then the partisan crowd on Court One will be in for a treat.
The Australian-born star looked set for a hammering when she capitulated in the first set, but Robson then launched a remarkable comeback to overcome her nerve-riddled opponent 1-6 7-5 6-3.
Robson had an off-day with her forehand - one of her biggest weapons - yesterday and it was clear to see early on that she was nervous.
She admits, therefore, that there is still room for improvement in her game.
"I can definitely play better than I did against Erakovic," the Londoner said.
"But it's tough to play your best tennis all the time.
"What I've been working on is just accepting that I'm not going to play great tennis in every match. I'm going to have to work on being consistent."