Update: July 2017.
Even though it doesn't happen a ton, this still is taking place and the behavior is different from posts. I've been meaning to get back to this but it fell into my large backlog of things to look at. I've now written it up internally for review in hopes it can be implemented without too much hassle. I'll change this to status-deferred while we move it through our process.
I've been meaning to look at this for a while to see how often users actually do this and to see if it's worth the effort to put these restrictions in place.
The stats are for the past year of comment flags on sites that had at least one comment flagged multiple times by the same user, including only the comment flags, not constructive, too chatty, rude, or obsolete. I purposely left out the "other" comment flag because that would be the one a user would be left with if this was implemented.
After pulling the numbers to see what percentage of comments are being flagged multiple times as you'll notice it doesn't happen a lot.
# Comment Flags # Re-flags % Reflagged Site Name
--------------- ---------- ----------- ---------
209406 969 0.46 Stack Overflow
3592 36 1.0 Super User
1041 14 1.34 Server Fault
21 2 9.52 Stack Apps
208 6 2.88 Home Improvement
231 2 0.87 Game Developers
1318 14 1.06 Gaming
124 4 3.23 GIS
8619 169 1.96 Mathematics
350 2 0.57 Photography
383 23 6.01 Statistical Analysis
408 2 0.49 Web Apps
117 2 1.71 Webmasters
594 41 6.9 Apple
2199 28 1.27 English Language and Usage
269 10 3.72 Personal Finance and Money
2768 11 0.4 Role-playing Games
1030 6 0.58 TeX - LaTeX
2103 26 1.24 Ubuntu
1078 6 0.56 Unix and Linux
156 6 3.85 WordPress
1892 28 1.48 Programmers
724 4 0.55 Android Enthusiasts
186 22 11.83 Board and Card Games
1479 8 0.54 Physics
1325 18 1.36 IT Security
47 2 4.26 Writers
761 24 3.15 Electronics and Robotics
267 6 2.25 Graphic Design
7590 60 0.79 Database Administrators
2578 49 1.9 Science Fiction
3813 29 0.76 Code Review
794 10 1.26 Code Golf
82 15 18.29 Quantitative Finance
779 51 6.55 Skeptics
450 4 0.89 Drupal Answers
133 8 6.02 Motor Vehicle Maintenance and Repair
438 2 0.46 Parenting
55 2 3.64 SharePoint
301 6 1.99 Musical Practice and Performance
36 2 5.56 Software Quality Assurance and Testing
1938 14 0.72 Jewish Life and Learning
603 19 3.15 German Language and Usage
219 6 2.74 Japanese Language and Usage
52 2 3.85 Gardening and Landscaping
332 12 3.61 Philosophy
902 29 3.22 Travel
41 2 4.88 Signal Processing
2148 6 0.28 Christianity
560 7 1.25 Movies
19 2 10.53 Chinese Language and Usage
401 5 1.25 Biology
133 2 1.5 The Great Outdoors
61 4 6.56 Martial Arts
1117 25 2.24 Academia
1765 10 0.57 Computer Science
1991 2 0.1 The Workplace
1114 6 0.54 Chemistry
36 6 16.67 Chess
965 26 2.69 Islam
100 2 2.0 Salesforce
257 2 0.78 User Experience
15 2 13.33 Audio-Video Production
131 12 9.16 Magento
414 16 3.86 English Language Learners
38 2 5.26 Sustainable Living
107 2 1.87 Network Engineering Stack Exchange
37 2 5.41 Open Data Stack Exchange
202 2 0.99 Blender Stack Exchange
596 6 1.01 MathOverflow
525 2 0.38 Space Exploration Stack Exchange
1119 8 0.71 Stack Overflow em Português
337 2 0.59 Aviation Stack Exchange
754 4 0.53 Software Recommendations Stack Exchange
43 2 4.65 Mathematics Educators Stack Exchange
672 10 1.49 Puzzling Stack Exchange
54 4 7.41 Buddhism Stack Exchange
186 6 3.23 Hinduism Stack Exchange
109 6 5.5 スタック・オーバーフロー
49 2 4.08 Emacs Stack Exchange
41 2 4.88 History of Science and Mathematics Stack Exchange
353 8 2.27 Lifehacks Stack Exchange
912 10 1.1 Engineering Stack Exchange
10 2 20.0 Coffee Stack Exchange
41 2 4.88 Music Fans Stack Exchange
2505 8 0.32 Stack Overflow на русском
254 4 1.57 Law Stack Exchange
113 4 3.54 Stack Overflow en español
The above numbers are when the same user flags the same comment multiple times with what could be different flags. Since you initially asked about users flagging the same comment multiple times with the same flag, I figured I better pull those stats as well:
# Comment Flags # Re-flags % Reflagged Site Name
--------------- ---------- ----------- ---------
209421 747 0.36 Stack Overflow
3579 27 0.75 Super User
1042 11 1.06 Server Fault
22 2 9.09 Stack Apps
210 4 1.9 Home Improvement
228 2 0.88 Game Developers
1316 14 1.06 Gaming
120 4 3.33 GIS
8625 133 1.54 Mathematics
349 2 0.57 Photography
386 17 4.4 Statistical Analysis
408 2 0.49 Web Apps
120 2 1.67 Webmasters
590 38 6.44 Apple
2232 19 0.85 English Language and Usage
263 6 2.28 Personal Finance and Money
2752 11 0.4 Role-playing Games
1045 6 0.57 TeX - LaTeX
2101 16 0.76 Ubuntu
1073 2 0.19 Unix and Linux
163 2 1.23 WordPress
1892 22 1.16 Programmers
726 4 0.55 Android Enthusiasts
187 20 10.7 Board and Card Games
1502 6 0.4 Physics
1327 18 1.36 IT Security
47 2 4.26 Writers
760 20 2.63 Electronics and Robotics
269 2 0.74 Graphic Design
7799 56 0.72 Database Administrators
2601 33 1.27 Science Fiction
3804 23 0.6 Code Review
798 6 0.75 Code Golf
82 15 18.29 Quantitative Finance
775 44 5.68 Skeptics
451 4 0.89 Drupal Answers
135 5 3.7 Motor Vehicle Maintenance and Repair
440 2 0.45 Parenting
55 2 3.64 SharePoint
300 4 1.33 Musical Practice and Performance
37 2 5.41 Software Quality Assurance and Testing
1951 8 0.41 Jewish Life and Learning
600 17 2.83 German Language and Usage
213 6 2.82 Japanese Language and Usage
50 2 4 Gardening and Landscaping
334 12 3.59 Philosophy
915 26 2.84 Travel
41 2 4.88 Signal Processing
2144 6 0.28 Christianity
563 2 0.36 Movies
19 2 10.53 Chinese Language and Usage
384 5 1.3 Biology
137 2 1.46 The Great Outdoors
61 4 6.56 Martial Arts
1075 17 1.58 Academia
1755 7 0.4 Computer Science
1116 2 0.18 Chemistry
36 6 16.67 Chess
968 16 1.65 Islam
102 2 1.96 Salesforce
250 2 0.8 User Experience
15 2 13.33 Audio-Video Production
131 12 9.16 Magento
409 14 3.42 English Language Learners
41 2 4.88 Sustainable Living
87 2 2.3 Network Engineering Stack Exchange
37 2 5.41 Open Data Stack Exchange
207 4 1.93 Blender Stack Exchange
598 4 0.67 MathOverflow
523 2 0.38 Space Exploration Stack Exchange
1119 8 0.71 Stack Overflow em Português
764 2 0.26 Software Recommendations Stack Exchange
742 6 0.81 Puzzling Stack Exchange
54 2 3.7 Buddhism Stack Exchange
177 6 3.39 Hinduism Stack Exchange
109 4 3.67 スタック・オーバーフロー
49 2 4.08 Emacs Stack Exchange
41 2 4.88 History of Science and Mathematics Stack Exchange
369 6 1.63 Lifehacks Stack Exchange
914 10 1.09 Engineering Stack Exchange
40 2 5 Music Fans Stack Exchange
2515 2 0.08 Stack Overflow на русском
266 3 1.13 Law Stack Exchange
114 4 3.51 Stack Overflow en español
Considering the percentage of comments re-flagged is so low, I'm not entirely sold on this idea but I'm not saying no...yet. Since it's more of an annoyance than a common problem, I'm going to status-review this to see how much work would be involved.