Key Inclusion Criteria:  Generally healthy males and females who are 12 years and older at the time of screening. Weight must be ≥40 kg at the time of screening  Documented or patient reported history (for at least 2 years) of symptomatic cat allergen-triggered allergic rhinitis with or without conjunctivitis and with or without asthma as defined by all of the following criteria:  Positive skin prick test (SPT) with cat hair extract (mean wheal diameter at least 5 mm greater than a negative control) at screening Positive allergen-specific IgE (sIgE) tests for cat hair and Fel d 1 (≥0.7 kUa/L at screening) Documented or patient reported history of nasal and/or ocular symptoms upon cat exposure Symptomatic despite the use of medications to treat their nasal and/or ocular symptoms At least 1 generally healthy cat (that is unlikely to die during the study) living in the home resulting in regular exposure A daily total rhinitis/conjunctivitis symptom score of at least 8/18 during at least 8 days of the 15-day baseline assessment period and use of standard, therapeutic doses of pharmacotherapy for the treatment of allergic rhinoconjunctivitis on at least 8 days of the 15-day baseline assessment period.  Key Exclusion Criteria:  History of significant multiple and/or severe allergies that would potentially interfere with the assessments during the baseline and 12-week efficacy assessment periods or confound results, per investigator discretion, including significant rhinitis or sinusitis due to daily contact with other allergens causing symptoms that are expected to coincide with the baseline period or any of the efficacy assessment periods Received REGN1908-1909 in a prior REGN1908-1909 clinical trial (receipt of placebo in a previous trial is allowed) Active lung disease other than asthma FEV1 less than 70% of predicted at screening or randomization Treatment with an investigational drug within 2 months or within 5 half-lives (if known), whichever is longer, prior to screening Persistent chronic or recurring acute infection requiring treatment with antibiotics, antivirals, or antifungals, or any untreated respiratory infections within 4 weeks prior to screening. Patients may be re-evaluated after resolution of symptoms and specified time duration  NOTE: Other protocol-defined Inclusion/ Exclusion Criteria apply