**Core Concept**
NSAIDs (Non-Steroidal Anti-Inflammatory Drugs) are a class of medications that primarily work by inhibiting the enzyme cyclooxygenase (COX), which is responsible for the production of prostaglandins. Prostaglandins play a crucial role in mediating inflammation, pain, and fever. By blocking COX, NSAIDs decrease the synthesis of prostaglandins, leading to their therapeutic effects.

**Why the Correct Answer is Right**
The correct answer will be a statement that is not true about NSAIDs. To determine this, we need to examine the properties and actions of NSAIDs. NSAIDs have analgesic, anti-inflammatory, and antipyretic effects, primarily through the inhibition of COX enzymes (COX-1 and COX-2). They also have a potential side effect of causing gastric mucosal damage due to the inhibition of COX-1, which protects the stomach lining.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is incorrect because NSAIDs do indeed have analgesic, anti-inflammatory, and antipyretic effects, primarily through the inhibition of COX enzymes.
* **Option B:** This option is incorrect because NSAIDs do have the potential side effect of causing gastric mucosal damage due to the inhibition of COX-1.
* **Option D:** This option is incorrect because NSAIDs do indeed have a ceiling effect, meaning that beyond a certain dose, the therapeutic effect does not increase further.

**Clinical Pearl / High-Yield Fact**
One important clinical consideration when prescribing NSAIDs is to be aware of the risk of gastrointestinal side effects, particularly in patients with a history of peptic ulcer disease or those taking anticoagulant medications. Patients should also be advised to take NSAIDs with food and to monitor for signs of gastrointestinal bleeding.

**Correct Answer: A.**